Calculus Examples

$\int \frac{3 x^{2} - 11}{x^{4} - 1} d x$

Step 1

Write the fraction using partial fraction decomposition.

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Step 1.1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1.1

Factor the fraction.

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Step 1.1.1.1

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{3 x^{2} - 11}{{(x^{2})}^{2} - 1}$

Step 1.1.1.2

Rewrite $1$ as $1^{2}$ .

$\frac{3 x^{2} - 11}{{(x^{2})}^{2} - 1^{2}}$

Step 1.1.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{2}$ and $b = 1$ .

$\frac{3 x^{2} - 11}{(x^{2} + 1) (x^{2} - 1)}$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

Rewrite $1$ as $1^{2}$ .

$\frac{3 x^{2} - 11}{(x^{2} + 1) (x^{2} - 1^{2})}$

Step 1.1.1.4.2

Factor.

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Step 1.1.1.4.2.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{3 x^{2} - 11}{(x^{2} + 1) ((x + 1) (x - 1))}$

Step 1.1.1.4.2.2

Remove unnecessary parentheses.

$\frac{3 x^{2} - 11}{(x^{2} + 1) (x + 1) (x - 1)}$

Step 1.1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor is 2nd order, $2$ terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.

$\frac{A x + B}{x^{2} + 1}$

Step 1.1.3

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $C$ .

$\frac{A x + B}{x^{2} + 1} + \frac{C}{x + 1}$

Step 1.1.4

$\frac{A x + B}{x^{2} + 1} + \frac{C}{x + 1} + \frac{D}{x - 1}$

Step 1.1.5

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x^{2} + 1) (x + 1) (x - 1)$ .

$\frac{(3 x^{2} - 11) (x^{2} + 1) (x + 1) (x - 1)}{(x^{2} + 1) (x + 1) (x - 1)} = \frac{(A x + B) (x^{2} + 1) (x + 1) (x - 1)}{x^{2} + 1} + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.6

Cancel the common factor of $x^{2} + 1$ .

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Step 1.1.6.1

Cancel the common factor.

Step 1.1.6.2

Rewrite the expression.

$\frac{((3 x^{2} - 11) (x + 1)) (x - 1)}{(x + 1) (x - 1)} = \frac{(A x + B) (x^{2} + 1) (x + 1) (x - 1)}{x^{2} + 1} + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.7

Cancel the common factor of $x + 1$ .

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Step 1.1.7.1

Cancel the common factor.

$\frac{(3 x^{2} - 11) (x + 1) (x - 1)}{(x + 1) (x - 1)} = \frac{(A x + B) (x^{2} + 1) (x + 1) (x - 1)}{x^{2} + 1} + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.7.2

Rewrite the expression.

$\frac{(3 x^{2} - 11) (x - 1)}{x - 1} = \frac{(A x + B) (x^{2} + 1) (x + 1) (x - 1)}{x^{2} + 1} + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.8

Cancel the common factor of $x - 1$ .

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Step 1.1.8.1

Cancel the common factor.

$\frac{(3 x^{2} - 11) (x - 1)}{x - 1} = \frac{(A x + B) (x^{2} + 1) (x + 1) (x - 1)}{x^{2} + 1} + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.8.2

Divide $3 x^{2} - 11$ by $1$ .

$3 x^{2} - 11 = \frac{(A x + B) (x^{2} + 1) (x + 1) (x - 1)}{x^{2} + 1} + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9

Simplify each term.

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Step 1.1.9.1

Cancel the common factor of $x^{2} + 1$ .

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Step 1.1.9.1.1

Cancel the common factor.

$3 x^{2} - 11 = \frac{(A x + B) (x^{2} + 1) (x + 1) (x - 1)}{x^{2} + 1} + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.1.2

Divide $((A x + B) (x + 1)) (x - 1)$ by $1$ .

$3 x^{2} - 11 = ((A x + B) (x + 1)) (x - 1) + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.2

Expand $(A x + B) (x + 1)$ using the FOIL Method.

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Step 1.1.9.2.1

Apply the distributive property.

$3 x^{2} - 11 = (A x (x + 1) + B (x + 1)) (x - 1) + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.2.2

Apply the distributive property.

$3 x^{2} - 11 = (A x \cdot x + A x \cdot 1 + B (x + 1)) (x - 1) + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.2.3

Apply the distributive property.

$3 x^{2} - 11 = (A x \cdot x + A x \cdot 1 + B x + B \cdot 1) (x - 1) + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.3

Simplify each term.

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Step 1.1.9.3.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.9.3.1.1

Move $x$ .

$3 x^{2} - 11 = (A (x \cdot x) + A x \cdot 1 + B x + B \cdot 1) (x - 1) + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.3.1.2

Multiply $x$ by $x$ .

$3 x^{2} - 11 = (A x^{2} + A x \cdot 1 + B x + B \cdot 1) (x - 1) + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.3.2

Multiply $A$ by $1$ .

$3 x^{2} - 11 = (A x^{2} + A x + B x + B \cdot 1) (x - 1) + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.3.3

Multiply $B$ by $1$ .

$3 x^{2} - 11 = (A x^{2} + A x + B x + B) (x - 1) + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.4

Expand $(A x^{2} + A x + B x + B) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

$3 x^{2} - 11 = A x^{2} x + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x + B x \cdot - 1 + B x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.5

Combine the opposite terms in $A x^{2} x + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x + B x \cdot - 1 + B x + B \cdot - 1$ .

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Step 1.1.9.5.1

Reorder the factors in the terms $B x \cdot - 1$ and $B x$ .

$3 x^{2} - 11 = A x^{2} x + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x - B x + B x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.5.2

Add $- B x$ and $B x$ .

$3 x^{2} - 11 = A x^{2} x + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x + 0 + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.5.3

Add $A x^{2} x + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x$ and $0$ .

$3 x^{2} - 11 = A x^{2} x + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6

Simplify each term.

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Step 1.1.9.6.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.9.6.1.1

Move $x$ .

$3 x^{2} - 11 = A (x \cdot x^{2}) + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.9.6.1.2.1

Raise $x$ to the power of $1$ .

Step 1.1.9.6.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{2} - 11 = A x^{1 + 2} + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.1.3

Add $1$ and $2$ .

$3 x^{2} - 11 = A x^{3} + A x^{2} \cdot - 1 + A x \cdot x + A x \cdot - 1 + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.2

Move $- 1$ to the left of $A x^{2}$ .

$3 x^{2} - 11 = A x^{3} - 1 \cdot (A x^{2}) + A x \cdot x + A x \cdot - 1 + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.3

Rewrite $- 1 (A x^{2})$ as $- (A x^{2})$ .

$3 x^{2} - 11 = A x^{3} - (A x^{2}) + A x \cdot x + A x \cdot - 1 + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.4

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.9.6.4.1

Move $x$ .

$3 x^{2} - 11 = A x^{3} - A x^{2} + A (x \cdot x) + A x \cdot - 1 + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.4.2

Multiply $x$ by $x$ .

$3 x^{2} - 11 = A x^{3} - A x^{2} + A x^{2} + A x \cdot - 1 + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.5

Move $- 1$ to the left of $A x$ .

$3 x^{2} - 11 = A x^{3} - A x^{2} + A x^{2} - 1 \cdot (A x) + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.6

Rewrite $- 1 (A x)$ as $- (A x)$ .

$3 x^{2} - 11 = A x^{3} - A x^{2} + A x^{2} - (A x) + B x \cdot x + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.7

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.9.6.7.1

Move $x$ .

$3 x^{2} - 11 = A x^{3} - A x^{2} + A x^{2} - A x + B (x \cdot x) + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.7.2

Multiply $x$ by $x$ .

$3 x^{2} - 11 = A x^{3} - A x^{2} + A x^{2} - A x + B x^{2} + B \cdot - 1 + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.8

Move $- 1$ to the left of $B$ .

$3 x^{2} - 11 = A x^{3} - A x^{2} + A x^{2} - A x + B x^{2} - 1 \cdot B + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.6.9

Rewrite $- 1 B$ as $- B$ .

$3 x^{2} - 11 = A x^{3} - A x^{2} + A x^{2} - A x + B x^{2} - B + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.7

Combine the opposite terms in $A x^{3} - A x^{2} + A x^{2} - A x + B x^{2} - B$ .

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Step 1.1.9.7.1

Add $- A x^{2}$ and $A x^{2}$ .

$3 x^{2} - 11 = A x^{3} + 0 - A x + B x^{2} - B + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.7.2

Add $A x^{3}$ and $0$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.8

Cancel the common factor of $x + 1$ .

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Step 1.1.9.8.1

Cancel the common factor.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + \frac{(C) (x^{2} + 1) (x + 1) (x - 1)}{x + 1} + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.8.2

Divide $(C) (x^{2} + 1) (x - 1)$ by $1$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + (C) (x^{2} + 1) (x - 1) + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.9

Apply the distributive property.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + (C x^{2} + C \cdot 1) (x - 1) + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.10

Multiply $C$ by $1$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + (C x^{2} + C) (x - 1) + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.11

Expand $(C x^{2} + C) (x - 1)$ using the FOIL Method.

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Step 1.1.9.11.1

Apply the distributive property.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{2} (x - 1) + C (x - 1) + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.11.2

Apply the distributive property.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{2} x + C x^{2} \cdot - 1 + C (x - 1) + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.11.3

Apply the distributive property.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{2} x + C x^{2} \cdot - 1 + C x + C \cdot - 1 + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.12

Simplify each term.

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Step 1.1.9.12.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.9.12.1.1

Move $x$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C (x \cdot x^{2}) + C x^{2} \cdot - 1 + C x + C \cdot - 1 + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.12.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.9.12.1.2.1

Raise $x$ to the power of $1$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C (x \cdot x^{2}) + C x^{2} \cdot - 1 + C x + C \cdot - 1 + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.12.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{1 + 2} + C x^{2} \cdot - 1 + C x + C \cdot - 1 + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.12.1.3

Add $1$ and $2$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} + C x^{2} \cdot - 1 + C x + C \cdot - 1 + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.12.2

Move $- 1$ to the left of $C x^{2}$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - 1 \cdot (C x^{2}) + C x + C \cdot - 1 + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.12.3

Rewrite $- 1 (C x^{2})$ as $- (C x^{2})$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - (C x^{2}) + C x + C \cdot - 1 + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.12.4

Move $- 1$ to the left of $C$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - 1 \cdot C + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.12.5

Rewrite $- 1 C$ as $- C$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.13

Cancel the common factor of $x - 1$ .

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Step 1.1.9.13.1

Cancel the common factor.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + \frac{(D) (x^{2} + 1) (x + 1) (x - 1)}{x - 1}$

Step 1.1.9.13.2

Divide $(D) (x^{2} + 1) (x + 1)$ by $1$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + (D) (x^{2} + 1) (x + 1)$

Step 1.1.9.14

Apply the distributive property.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + (D x^{2} + D \cdot 1) (x + 1)$

Step 1.1.9.15

Multiply $D$ by $1$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + (D x^{2} + D) (x + 1)$

Step 1.1.9.16

Expand $(D x^{2} + D) (x + 1)$ using the FOIL Method.

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Step 1.1.9.16.1

Apply the distributive property.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{2} (x + 1) + D (x + 1)$

Step 1.1.9.16.2

Apply the distributive property.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{2} x + D x^{2} \cdot 1 + D (x + 1)$

Step 1.1.9.16.3

Apply the distributive property.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{2} x + D x^{2} \cdot 1 + D x + D \cdot 1$

Step 1.1.9.17

Simplify each term.

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Step 1.1.9.17.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.9.17.1.1

Move $x$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D (x \cdot x^{2}) + D x^{2} \cdot 1 + D x + D \cdot 1$

Step 1.1.9.17.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.9.17.1.2.1

Raise $x$ to the power of $1$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D (x \cdot x^{2}) + D x^{2} \cdot 1 + D x + D \cdot 1$

Step 1.1.9.17.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{1 + 2} + D x^{2} \cdot 1 + D x + D \cdot 1$

Step 1.1.9.17.1.3

Add $1$ and $2$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{3} + D x^{2} \cdot 1 + D x + D \cdot 1$

Step 1.1.9.17.2

Multiply $D$ by $1$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{3} + D x^{2} + D x + D \cdot 1$

Step 1.1.9.17.3

Multiply $D$ by $1$ .

$3 x^{2} - 11 = A x^{3} - A x + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{3} + D x^{2} + D x + D$

Step 1.1.10

Simplify the expression.

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Step 1.1.10.1

Move $A$ .

$3 x^{2} - 11 = A x^{3} - 1 x A + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{3} + D x^{2} + D x + D$

Step 1.1.10.2

Reorder $B$ and $x^{2}$ .

$3 x^{2} - 11 = A x^{3} - 1 x A + B x^{2} - B + C x^{3} - C x^{2} + C x - C + D x^{3} + D x^{2} + D x + D$

Step 1.1.10.3

Move $- C$ .

$3 x^{2} - 11 = A x^{3} - 1 x A + B x^{2} - B + C x^{3} - C x^{2} + C x + D x^{3} + D x^{2} + D x - C + D$

Step 1.1.10.4

Move $- B$ .

$3 x^{2} - 11 = A x^{3} - 1 x A + B x^{2} + C x^{3} - C x^{2} + C x + D x^{3} + D x^{2} + D x - B - C + D$

Step 1.1.10.5

Move $C x$ .

$3 x^{2} - 11 = A x^{3} - 1 x A + B x^{2} + C x^{3} - C x^{2} + D x^{3} + D x^{2} + C x + D x - B - C + D$

Step 1.1.10.6

Move $- 1 x A$ .

$3 x^{2} - 11 = A x^{3} + B x^{2} + C x^{3} - C x^{2} + D x^{3} + D x^{2} - 1 x A + C x + D x - B - C + D$

Step 1.1.10.7

Move $- C x^{2}$ .

$3 x^{2} - 11 = A x^{3} + B x^{2} + C x^{3} + D x^{3} - C x^{2} + D x^{2} - 1 x A + C x + D x - B - C + D$

Step 1.1.10.8

Move $B x^{2}$ .

$3 x^{2} - 11 = A x^{3} + C x^{3} + D x^{3} + B x^{2} - C x^{2} + D x^{2} - 1 x A + C x + D x - B - C + D$

Step 1.2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + C + D$

Step 1.2.2

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$3 = B - 1 C + D$

Step 1.2.3

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 1 A + C + D$

Step 1.2.4

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$- 11 = - 1 B - 1 C + D$

Step 1.2.5

Set up the system of equations to find the coefficients of the partial fractions.

$0 = A + C + D$

$3 = B - 1 C + D$

$0 = - 1 A + C + D$

$- 11 = - 1 B - 1 C + D$

$0 = A + C + D$

$3 = B - 1 C + D$

$0 = - 1 A + C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Solve for $A$ in $0 = A + C + D$ .

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Step 1.3.1.1

Rewrite the equation as $A + C + D = 0$ .

$A + C + D = 0$

$3 = B - 1 C + D$

$0 = - 1 A + C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.1.2

Move all terms not containing $A$ to the right side of the equation.

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Step 1.3.1.2.1

Subtract $C$ from both sides of the equation.

$A + D = - C$

$3 = B - 1 C + D$

$0 = - 1 A + C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.1.2.2

Subtract $D$ from both sides of the equation.

$A = - C - D$

$3 = B - 1 C + D$

$0 = - 1 A + C + D$

$- 11 = - 1 B - 1 C + D$

$A = - C - D$

$3 = B - 1 C + D$

$0 = - 1 A + C + D$

$- 11 = - 1 B - 1 C + D$

$A = - C - D$

$3 = B - 1 C + D$

$0 = - 1 A + C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.2

Replace all occurrences of $A$ with $- C - D$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $A$ in $0 = - 1 A + C + D$ with $- C - D$ .

$0 = - 1 (- C - D) + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify $- 1 (- C - D) + C + D$ .

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Step 1.3.2.2.1.1

Simplify each term.

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Step 1.3.2.2.1.1.1

Apply the distributive property.

$0 = - 1 (- C) - 1 (- D) + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.2.2.1.1.2

Multiply $- 1 (- C)$ .

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Step 1.3.2.2.1.1.2.1

Multiply $- 1$ by $- 1$ .

$0 = 1 C - 1 (- D) + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.2.2.1.1.2.2

Multiply $C$ by $1$ .

$0 = C - 1 (- D) + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$0 = C - 1 (- D) + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.2.2.1.1.3

Multiply $- 1 (- D)$ .

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Step 1.3.2.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$0 = C + 1 D + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.2.2.1.1.3.2

Multiply $D$ by $1$ .

$0 = C + D + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$0 = C + D + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$0 = C + D + C + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.2.2.1.2

Simplify by adding terms.

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Step 1.3.2.2.1.2.1

Add $C$ and $C$ .

$0 = 2 C + D + D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.2.2.1.2.2

Add $D$ and $D$ .

$0 = 2 C + 2 D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$0 = 2 C + 2 D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$0 = 2 C + 2 D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$0 = 2 C + 2 D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$0 = 2 C + 2 D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3

Solve for $C$ in $0 = 2 C + 2 D$ .

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Step 1.3.3.1

Rewrite the equation as $2 C + 2 D = 0$ .

$2 C + 2 D = 0$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.2

Subtract $2 D$ from both sides of the equation.

$2 C = - 2 D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.3

Divide each term in $2 C = - 2 D$ by $2$ and simplify.

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Step 1.3.3.3.1

Divide each term in $2 C = - 2 D$ by $2$ .

$\frac{2 C}{2} = \frac{- 2 D}{2}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.3.2

Simplify the left side.

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Step 1.3.3.3.2.1

Cancel the common factor of $2$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{2 C}{2} = \frac{- 2 D}{2}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{- 2 D}{2}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$C = \frac{- 2 D}{2}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$C = \frac{- 2 D}{2}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Cancel the common factor of $- 2$ and $2$ .

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Step 1.3.3.3.3.1.1

Factor $2$ out of $- 2 D$ .

$C = \frac{2 (- D)}{2}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.3.3.1.2

Cancel the common factors.

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Step 1.3.3.3.3.1.2.1

Factor $2$ out of $2$ .

$C = \frac{2 (- D)}{2 (1)}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.3.3.1.2.2

Cancel the common factor.

$C = \frac{2 (- D)}{2 \cdot 1}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.3.3.1.2.3

Rewrite the expression.

$C = \frac{- D}{1}$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.3.3.3.1.2.4

Divide $- D$ by $1$ .

$C = - D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$C = - D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$C = - D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$C = - D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$C = - D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$C = - D$

$A = - C - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4

Replace all occurrences of $C$ with $- D$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $C$ in $A = - C - D$ with $- D$ .

$A = - (- D) - D$

$C = - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify $- (- D) - D$ .

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Step 1.3.4.2.1.1

Multiply $- (- D)$ .

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Step 1.3.4.2.1.1.1

Multiply $- 1$ by $- 1$ .

$A = 1 D - D$

$C = - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4.2.1.1.2

Multiply $D$ by $1$ .

$A = D - D$

$C = - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$A = D - D$

$C = - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4.2.1.2

Subtract $D$ from $D$ .

$A = 0$

$C = - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$A = 0$

$C = - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

$A = 0$

$C = - D$

$3 = B - 1 C + D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4.3

Replace all occurrences of $C$ in $3 = B - 1 C + D$ with $- D$ .

$3 = B - 1 (- D) + D$

$A = 0$

$C = - D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4.4

Simplify the right side.

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Step 1.3.4.4.1

Simplify $B - 1 (- D) + D$ .

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Step 1.3.4.4.1.1

Multiply $- 1 (- D)$ .

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Step 1.3.4.4.1.1.1

Multiply $- 1$ by $- 1$ .

$3 = B + 1 D + D$

$A = 0$

$C = - D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4.4.1.1.2

Multiply $D$ by $1$ .

$3 = B + D + D$

$A = 0$

$C = - D$

$- 11 = - 1 B - 1 C + D$

$3 = B + D + D$

$A = 0$

$C = - D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4.4.1.2

Add $D$ and $D$ .

$3 = B + 2 D$

$A = 0$

$C = - D$

$- 11 = - 1 B - 1 C + D$

$3 = B + 2 D$

$A = 0$

$C = - D$

$- 11 = - 1 B - 1 C + D$

$3 = B + 2 D$

$A = 0$

$C = - D$

$- 11 = - 1 B - 1 C + D$

Step 1.3.4.5

Replace all occurrences of $C$ in $- 11 = - 1 B - 1 C + D$ with $- D$ .

$- 11 = - 1 B - 1 (- D) + D$

$3 = B + 2 D$

$A = 0$

$C = - D$

Step 1.3.4.6

Simplify the right side.

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Step 1.3.4.6.1

Simplify $- 1 B - 1 (- D) + D$ .

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Step 1.3.4.6.1.1

Simplify each term.

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Step 1.3.4.6.1.1.1

Rewrite $- 1 B$ as $- B$ .

$- 11 = - B - 1 (- D) + D$

$3 = B + 2 D$

$A = 0$

$C = - D$

Step 1.3.4.6.1.1.2

Multiply $- 1 (- D)$ .

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Step 1.3.4.6.1.1.2.1

Multiply $- 1$ by $- 1$ .

$- 11 = - B + 1 D + D$

$3 = B + 2 D$

$A = 0$

$C = - D$

Step 1.3.4.6.1.1.2.2

Multiply $D$ by $1$ .

$- 11 = - B + D + D$

$3 = B + 2 D$

$A = 0$

$C = - D$

$- 11 = - B + D + D$

$3 = B + 2 D$

$A = 0$

$C = - D$

$- 11 = - B + D + D$

$3 = B + 2 D$

$A = 0$

$C = - D$

Step 1.3.4.6.1.2

Add $D$ and $D$ .

$- 11 = - B + 2 D$

$3 = B + 2 D$

$A = 0$

$C = - D$

$- 11 = - B + 2 D$

$3 = B + 2 D$

$A = 0$

$C = - D$

$- 11 = - B + 2 D$

$3 = B + 2 D$

$A = 0$

$C = - D$

$- 11 = - B + 2 D$

$3 = B + 2 D$

$A = 0$

$C = - D$

Step 1.3.5

Solve for $B$ in $3 = B + 2 D$ .

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Step 1.3.5.1

Rewrite the equation as $B + 2 D = 3$ .

$B + 2 D = 3$

$- 11 = - B + 2 D$

$A = 0$

$C = - D$

Step 1.3.5.2

Subtract $2 D$ from both sides of the equation.

$B = 3 - 2 D$

$- 11 = - B + 2 D$

$A = 0$

$C = - D$

$B = 3 - 2 D$

$- 11 = - B + 2 D$

$A = 0$

$C = - D$

Step 1.3.6

Replace all occurrences of $B$ with $3 - 2 D$ in each equation.

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Step 1.3.6.1

Replace all occurrences of $B$ in $- 11 = - B + 2 D$ with $3 - 2 D$ .

$- 11 = - (3 - 2 D) + 2 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.6.2

Simplify the right side.

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Step 1.3.6.2.1

Simplify $- (3 - 2 D) + 2 D$ .

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Step 1.3.6.2.1.1

Simplify each term.

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Step 1.3.6.2.1.1.1

Apply the distributive property.

$- 11 = - 1 \cdot 3 - (- 2 D) + 2 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.6.2.1.1.2

Multiply $- 1$ by $3$ .

$- 11 = - 3 - (- 2 D) + 2 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.6.2.1.1.3

Multiply $- 2$ by $- 1$ .

$- 11 = - 3 + 2 D + 2 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$- 11 = - 3 + 2 D + 2 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.6.2.1.2

Add $2 D$ and $2 D$ .

$- 11 = - 3 + 4 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$- 11 = - 3 + 4 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$- 11 = - 3 + 4 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$- 11 = - 3 + 4 D$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.7

Solve for $D$ in $- 11 = - 3 + 4 D$ .

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Step 1.3.7.1

Rewrite the equation as $- 3 + 4 D = - 11$ .

$- 3 + 4 D = - 11$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.7.2

Move all terms not containing $D$ to the right side of the equation.

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Step 1.3.7.2.1

Add $3$ to both sides of the equation.

$4 D = - 11 + 3$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.7.2.2

Add $- 11$ and $3$ .

$4 D = - 8$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$4 D = - 8$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.7.3

Divide each term in $4 D = - 8$ by $4$ and simplify.

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Step 1.3.7.3.1

Divide each term in $4 D = - 8$ by $4$ .

$\frac{4 D}{4} = \frac{- 8}{4}$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.7.3.2

Simplify the left side.

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Step 1.3.7.3.2.1

Cancel the common factor of $4$ .

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Step 1.3.7.3.2.1.1

Cancel the common factor.

$\frac{4 D}{4} = \frac{- 8}{4}$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.7.3.2.1.2

Divide $D$ by $1$ .

$D = \frac{- 8}{4}$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$D = \frac{- 8}{4}$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$D = \frac{- 8}{4}$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.7.3.3

Simplify the right side.

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Step 1.3.7.3.3.1

Divide $- 8$ by $4$ .

$D = - 2$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$D = - 2$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$D = - 2$

$B = 3 - 2 D$

$A = 0$

$C = - D$

$D = - 2$

$B = 3 - 2 D$

$A = 0$

$C = - D$

Step 1.3.8

Replace all occurrences of $D$ with $- 2$ in each equation.

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Step 1.3.8.1

Replace all occurrences of $D$ in $B = 3 - 2 D$ with $- 2$ .

$B = 3 - 2 \cdot - 2$

$D = - 2$

$A = 0$

$C = - D$

Step 1.3.8.2

Simplify the right side.

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Step 1.3.8.2.1

Simplify $3 - 2 \cdot - 2$ .

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Step 1.3.8.2.1.1

Multiply $- 2$ by $- 2$ .

$B = 3 + 4$

$D = - 2$

$A = 0$

$C = - D$

Step 1.3.8.2.1.2

Add $3$ and $4$ .

$B = 7$

$D = - 2$

$A = 0$

$C = - D$

$B = 7$

$D = - 2$

$A = 0$

$C = - D$

$B = 7$

$D = - 2$

$A = 0$

$C = - D$

Step 1.3.8.3

Replace all occurrences of $D$ in $C = - D$ with $- 2$ .

$C = - (- 2)$

$B = 7$

$D = - 2$

$A = 0$

Step 1.3.8.4

Simplify the right side.

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Step 1.3.8.4.1

Multiply $- 1$ by $- 2$ .

$C = 2$

$B = 7$

$D = - 2$

$A = 0$

$C = 2$

$B = 7$

$D = - 2$

$A = 0$

$C = 2$

$B = 7$

$D = - 2$

$A = 0$

Step 1.3.9

List all of the solutions.

$C = 2, B = 7, D = - 2, A = 0$

Step 1.4

Replace each of the partial fraction coefficients in $\frac{A x + B}{x^{2} + 1} + \frac{C}{x + 1} + \frac{D}{x - 1}$ with the values found for $A$ , $B$ , $C$ , and $D$ .

$\frac{0 x + 7}{x^{2} + 1} + \frac{2}{x + 1} + \frac{- 2}{x - 1}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Factor $7$ out of $0 \cdot x + 7$ .

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Step 1.5.1.1.1

Factor $7$ out of $0 \cdot x$ .

$\frac{7 (0 \cdot x) + 7}{x^{2} + 1} + \frac{2}{x + 1} + \frac{- 2}{x - 1}$

Step 1.5.1.1.2

Factor $7$ out of $7$ .

$\frac{7 (0 \cdot x) + 7 (1)}{x^{2} + 1} + \frac{2}{x + 1} + \frac{- 2}{x - 1}$

Step 1.5.1.1.3

Factor $7$ out of $7 (0 \cdot x) + 7 (1)$ .

$\frac{7 (0 \cdot x + 1)}{x^{2} + 1} + \frac{2}{x + 1} + \frac{- 2}{x - 1}$

Step 1.5.1.2

Multiply $0$ by $x$ .

$\frac{7 (0 + 1)}{x^{2} + 1} + \frac{2}{x + 1} + \frac{- 2}{x - 1}$

Step 1.5.1.3

Add $0$ and $1$ .

$\frac{7 \cdot 1}{x^{2} + 1} + \frac{2}{x + 1} + \frac{- 2}{x - 1}$

Step 1.5.2

Multiply $7$ by $1$ .

$\frac{7}{x^{2} + 1} + \frac{2}{x + 1} + \frac{- 2}{x - 1}$

Step 1.5.3

Move the negative in front of the fraction.

$\int \frac{7}{x^{2} + 1} + \frac{2}{x + 1} - \frac{2}{x - 1} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{7}{x^{2} + 1} d x + \int \frac{2}{x + 1} d x + \int - \frac{2}{x - 1} d x$

Step 3

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int \frac{1}{x^{2} + 1} d x + \int \frac{2}{x + 1} d x + \int - \frac{2}{x - 1} d x$

Step 4

Simplify the expression.

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Step 4.1

Reorder $x^{2}$ and $1$ .

$7 \int \frac{1}{1 + x^{2}} d x + \int \frac{2}{x + 1} d x + \int - \frac{2}{x - 1} d x$

Step 4.2

Rewrite $1$ as $1^{2}$ .

$7 \int \frac{1}{1^{2} + x^{2}} d x + \int \frac{2}{x + 1} d x + \int - \frac{2}{x - 1} d x$

Step 5

The integral of $\frac{1}{1^{2} + x^{2}}$ with respect to $x$ is $\arctan (x) + C$ .

$7 (\arctan (x) + C) + \int \frac{2}{x + 1} d x + \int - \frac{2}{x - 1} d x$

Step 6

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$7 (\arctan (x) + C) + 2 \int \frac{1}{x + 1} d x + \int - \frac{2}{x - 1} d x$

Step 7

Let $u_{1} = x + 1$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 7.1

Let $u_{1} = x + 1$ . Find $\frac{d u_{1}}{d x}$ .

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Step 7.1.1

Differentiate $x + 1$ .

$\frac{d}{d x} [x + 1]$

Step 7.1.2

By the Sum Rule, the derivative of $x + 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 7.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [1]$

Step 7.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$1 + 0$

Step 7.1.5

Add $1$ and $0$ .

$1$

Step 7.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$7 (\arctan (x) + C) + 2 \int \frac{1}{u_{1}} d u_{1} + \int - \frac{2}{x - 1} d x$

Step 8

The integral of $\frac{1}{u_{1}}$ with respect to $u_{1}$ is $\ln (| u_{1} |)$ .

$7 (\arctan (x) + C) + 2 (\ln (| u_{1} |) + C) + \int - \frac{2}{x - 1} d x$

Step 9

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$7 (\arctan (x) + C) + 2 (\ln (| u_{1} |) + C) - \int \frac{2}{x - 1} d x$

Step 10

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$7 (\arctan (x) + C) + 2 (\ln (| u_{1} |) + C) - (2 \int \frac{1}{x - 1} d x)$

Step 11

Multiply $2$ by $- 1$ .

$7 (\arctan (x) + C) + 2 (\ln (| u_{1} |) + C) - 2 \int \frac{1}{x - 1} d x$

Step 12

Let $u_{2} = x - 1$ . Then $d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 12.1

Let $u_{2} = x - 1$ . Find $\frac{d u_{2}}{d x}$ .

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Step 12.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 12.1.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 12.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 1]$

Step 12.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 12.1.5

Add $1$ and $0$ .

$1$

Step 12.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$7 (\arctan (x) + C) + 2 (\ln (| u_{1} |) + C) - 2 \int \frac{1}{u_{2}} d u_{2}$

Step 13

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$7 (\arctan (x) + C) + 2 (\ln (| u_{1} |) + C) - 2 (\ln (| u_{2} |) + C)$

Step 14

Simplify.

$7 \arctan (x) + 2 \ln (| u_{1} |) - 2 \ln (| u_{2} |) + C$

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

Replace all occurrences of $u_{1}$ with $x + 1$ .

$7 \arctan (x) + 2 \ln (| x + 1 |) - 2 \ln (| u_{2} |) + C$

Step 15.2

Replace all occurrences of $u_{2}$ with $x - 1$ .

$7 \arctan (x) + 2 \ln (| x + 1 |) - 2 \ln (| x - 1 |) + C$