Calculus Examples

$\int \frac{2 x^{2} - 12 x + 4}{x^{3} - 4 x^{2}} d x$

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

Factor $2$ out of $2 x^{2} - 12 x + 4$ .

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Step 1.1.1.1.1

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2}) - 12 x + 4}{x^{3} - 4 x^{2}}$

Step 1.1.1.1.2

Factor $2$ out of $- 12 x$ .

$\frac{2 (x^{2}) + 2 (- 6 x) + 4}{x^{3} - 4 x^{2}}$

Step 1.1.1.1.3

Factor $2$ out of $4$ .

$\frac{2 x^{2} + 2 (- 6 x) + 2 \cdot 2}{x^{3} - 4 x^{2}}$

Step 1.1.1.1.4

Factor $2$ out of $2 x^{2} + 2 (- 6 x)$ .

$\frac{2 (x^{2} - 6 x) + 2 \cdot 2}{x^{3} - 4 x^{2}}$

Step 1.1.1.1.5

Factor $2$ out of $2 (x^{2} - 6 x) + 2 \cdot 2$ .

$\frac{2 (x^{2} - 6 x + 2)}{x^{3} - 4 x^{2}}$

Step 1.1.1.2

Factor $x^{2}$ out of $x^{3} - 4 x^{2}$ .

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Step 1.1.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

$\frac{2 (x^{2} - 6 x + 2)}{x^{2} x - 4 x^{2}}$

Step 1.1.1.2.2

Factor $x^{2}$ out of $- 4 x^{2}$ .

$\frac{2 (x^{2} - 6 x + 2)}{x^{2} x + x^{2} \cdot - 4}$

Step 1.1.1.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 4$ .

$\frac{2 (x^{2} - 6 x + 2)}{x^{2} (x - 4)}$

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 in the denominator is linear, put a single variable in its place $C$ .

$\frac{A}{x^{2}} + \frac{B}{x} + \frac{C}{x - 4}$

Step 1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $x^{2} (x - 4)$ .

$\frac{2 (x^{2} - 6 x + 2) (x^{2} (x - 4))}{x^{2} (x - 4)} = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.4

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

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Step 1.1.4.1

Cancel the common factor.

$\frac{2 (x^{2} - 6 x + 2) (x^{2} (x - 4))}{x^{2} (x - 4)} = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.4.2

Rewrite the expression.

$\frac{2 (x^{2} - 6 x + 2) (x - 4)}{x - 4} = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.5

Cancel the common factor of $x - 4$ .

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Step 1.1.5.1

Cancel the common factor.

$\frac{2 (x^{2} - 6 x + 2) (x - 4)}{x - 4} = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.5.2

Divide $2 (x^{2} - 6 x + 2)$ by $1$ .

$2 (x^{2} - 6 x + 2) = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.6

Apply the distributive property.

$2 x^{2} + 2 (- 6 x) + 2 \cdot 2 = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.7

Simplify.

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Step 1.1.7.1

Multiply $- 6$ by $2$ .

$2 x^{2} - 12 x + 2 \cdot 2 = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.7.2

Multiply $2$ by $2$ .

$2 x^{2} - 12 x + 4 = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8

Simplify each term.

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Step 1.1.8.1

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

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Step 1.1.8.1.1

Cancel the common factor.

$2 x^{2} - 12 x + 4 = \frac{A (x^{2} (x - 4))}{x^{2}} + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.1.2

Divide $A (x - 4)$ by $1$ .

$2 x^{2} - 12 x + 4 = A (x - 4) + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.2

Apply the distributive property.

$2 x^{2} - 12 x + 4 = A x + A \cdot - 4 + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.3

Move $- 4$ to the left of $A$ .

$2 x^{2} - 12 x + 4 = A x - 4 \cdot A + \frac{B (x^{2} (x - 4))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.4

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

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Step 1.1.8.4.1

Factor $x$ out of $B (x^{2} (x - 4))$ .

$2 x^{2} - 12 x + 4 = A x - 4 A + \frac{x (B (x (x - 4)))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.4.2

Cancel the common factors.

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Step 1.1.8.4.2.1

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

$2 x^{2} - 12 x + 4 = A x - 4 A + \frac{x (B (x (x - 4)))}{x} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.4.2.2

Factor $x$ out of $x^{1}$ .

$2 x^{2} - 12 x + 4 = A x - 4 A + \frac{x (B (x (x - 4)))}{x \cdot 1} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.4.2.3

Cancel the common factor.

$2 x^{2} - 12 x + 4 = A x - 4 A + \frac{x (B (x (x - 4)))}{x \cdot 1} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.4.2.4

Rewrite the expression.

$2 x^{2} - 12 x + 4 = A x - 4 A + \frac{B (x (x - 4))}{1} + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.4.2.5

Divide $B (x (x - 4))$ by $1$ .

$2 x^{2} - 12 x + 4 = A x - 4 A + B (x (x - 4)) + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.5

Apply the distributive property.

$2 x^{2} - 12 x + 4 = A x - 4 A + B (x \cdot x + x \cdot - 4) + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.6

Multiply $x$ by $x$ .

$2 x^{2} - 12 x + 4 = A x - 4 A + B (x^{2} + x \cdot - 4) + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.7

Move $- 4$ to the left of $x$ .

$2 x^{2} - 12 x + 4 = A x - 4 A + B (x^{2} - 4 \cdot x) + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.8

Apply the distributive property.

$2 x^{2} - 12 x + 4 = A x - 4 A + B x^{2} + B (- 4 x) + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.9

Rewrite using the commutative property of multiplication.

$2 x^{2} - 12 x + 4 = A x - 4 A + B x^{2} - 4 B x + \frac{(C) (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.10

Cancel the common factor of $x - 4$ .

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Step 1.1.8.10.1

Cancel the common factor.

$2 x^{2} - 12 x + 4 = A x - 4 A + B x^{2} - 4 B x + \frac{C (x^{2} (x - 4))}{x - 4}$

Step 1.1.8.10.2

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

$2 x^{2} - 12 x + 4 = A x - 4 A + B x^{2} - 4 B x + (C) (x^{2})$

$2 x^{2} - 12 x + 4 = A x - 4 A + B x^{2} - 4 B x + C x^{2}$

Step 1.1.9

Simplify the expression.

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Step 1.1.9.1

Move $B$ .

$2 x^{2} - 12 x + 4 = A x - 4 A + B x^{2} - 4 x B + C x^{2}$

Step 1.1.9.2

Move $- 4 A$ .

$2 x^{2} - 12 x + 4 = A x + B x^{2} - 4 x B + C x^{2} - 4 A$

Step 1.1.9.3

Move $- 4 x B$ .

$2 x^{2} - 12 x + 4 = A x + B x^{2} + C x^{2} - 4 x B - 4 A$

Step 1.1.9.4

Move $A x$ .

$2 x^{2} - 12 x + 4 = B x^{2} + C x^{2} + A x - 4 x B - 4 A$

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 $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$2 = B + C$

Step 1.2.2

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.

$- 12 = A - 4 B$

Step 1.2.3

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.

$4 = - 4 A$

Step 1.2.4

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

$2 = B + C$

$- 12 = A - 4 B$

$4 = - 4 A$

$2 = B + C$

$- 12 = A - 4 B$

$4 = - 4 A$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Solve for $A$ in $4 = - 4 A$ .

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Step 1.3.1.1

Rewrite the equation as $- 4 A = 4$ .

$- 4 A = 4$

$2 = B + C$

$- 12 = A - 4 B$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Divide each term in $- 4 A = 4$ by $- 4$ .

$\frac{- 4 A}{- 4} = \frac{4}{- 4}$

$2 = B + C$

$- 12 = A - 4 B$

Step 1.3.1.2.2

Simplify the left side.

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Step 1.3.1.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.3.1.2.2.1.1

Cancel the common factor.

$\frac{- 4 A}{- 4} = \frac{4}{- 4}$

$2 = B + C$

$- 12 = A - 4 B$

Step 1.3.1.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{4}{- 4}$

$2 = B + C$

$- 12 = A - 4 B$

$A = \frac{4}{- 4}$

$2 = B + C$

$- 12 = A - 4 B$

$A = \frac{4}{- 4}$

$2 = B + C$

$- 12 = A - 4 B$

Step 1.3.1.2.3

Simplify the right side.

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Step 1.3.1.2.3.1

Divide $4$ by $- 4$ .

$A = - 1$

$2 = B + C$

$- 12 = A - 4 B$

$A = - 1$

$2 = B + C$

$- 12 = A - 4 B$

$A = - 1$

$2 = B + C$

$- 12 = A - 4 B$

$A = - 1$

$2 = B + C$

$- 12 = A - 4 B$

Step 1.3.2

Replace all occurrences of $A$ with $- 1$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $A$ in $- 12 = A - 4 B$ with $- 1$ .

$- 12 = (- 1) - 4 B$

$A = - 1$

$2 = B + C$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Remove parentheses.

$- 12 = - 1 - 4 B$

$A = - 1$

$2 = B + C$

$- 12 = - 1 - 4 B$

$A = - 1$

$2 = B + C$

$- 12 = - 1 - 4 B$

$A = - 1$

$2 = B + C$

Step 1.3.3

Solve for $B$ in $- 12 = - 1 - 4 B$ .

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Step 1.3.3.1

Rewrite the equation as $- 1 - 4 B = - 12$ .

$- 1 - 4 B = - 12$

$A = - 1$

$2 = B + C$

Step 1.3.3.2

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

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Step 1.3.3.2.1

Add $1$ to both sides of the equation.

$- 4 B = - 12 + 1$

$A = - 1$

$2 = B + C$

Step 1.3.3.2.2

Add $- 12$ and $1$ .

$- 4 B = - 11$

$A = - 1$

$2 = B + C$

$- 4 B = - 11$

$A = - 1$

$2 = B + C$

Step 1.3.3.3

Divide each term in $- 4 B = - 11$ by $- 4$ and simplify.

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Step 1.3.3.3.1

Divide each term in $- 4 B = - 11$ by $- 4$ .

$\frac{- 4 B}{- 4} = \frac{- 11}{- 4}$

$A = - 1$

$2 = B + C$

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 $- 4$ .

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Step 1.3.3.3.2.1.1

Cancel the common factor.

$\frac{- 4 B}{- 4} = \frac{- 11}{- 4}$

$A = - 1$

$2 = B + C$

Step 1.3.3.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 11}{- 4}$

$A = - 1$

$2 = B + C$

$B = \frac{- 11}{- 4}$

$A = - 1$

$2 = B + C$

$B = \frac{- 11}{- 4}$

$A = - 1$

$2 = B + C$

Step 1.3.3.3.3

Simplify the right side.

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Step 1.3.3.3.3.1

Dividing two negative values results in a positive value.

$B = \frac{11}{4}$

$A = - 1$

$2 = B + C$

$B = \frac{11}{4}$

$A = - 1$

$2 = B + C$

$B = \frac{11}{4}$

$A = - 1$

$2 = B + C$

$B = \frac{11}{4}$

$A = - 1$

$2 = B + C$

Step 1.3.4

Replace all occurrences of $B$ with $\frac{11}{4}$ in each equation.

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Step 1.3.4.1

Replace all occurrences of $B$ in $2 = B + C$ with $\frac{11}{4}$ .

$2 = (\frac{11}{4}) + C$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Remove parentheses.

$2 = \frac{11}{4} + C$

$B = \frac{11}{4}$

$A = - 1$

$2 = \frac{11}{4} + C$

$B = \frac{11}{4}$

$A = - 1$

$2 = \frac{11}{4} + C$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.5

Solve for $C$ in $2 = \frac{11}{4} + C$ .

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Step 1.3.5.1

Rewrite the equation as $\frac{11}{4} + C = 2$ .

$\frac{11}{4} + C = 2$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.5.2

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

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Step 1.3.5.2.1

Subtract $\frac{11}{4}$ from both sides of the equation.

$C = 2 - \frac{11}{4}$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.5.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$C = 2 \cdot \frac{4}{4} - \frac{11}{4}$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.5.2.3

Combine $2$ and $\frac{4}{4}$ .

$C = \frac{2 \cdot 4}{4} - \frac{11}{4}$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.5.2.4

Combine the numerators over the common denominator.

$C = \frac{2 \cdot 4 - 11}{4}$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.5.2.5

Simplify the numerator.

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Step 1.3.5.2.5.1

Multiply $2$ by $4$ .

$C = \frac{8 - 11}{4}$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.5.2.5.2

Subtract $11$ from $8$ .

$C = \frac{- 3}{4}$

$B = \frac{11}{4}$

$A = - 1$

$C = \frac{- 3}{4}$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.5.2.6

Move the negative in front of the fraction.

$C = - \frac{3}{4}$

$B = \frac{11}{4}$

$A = - 1$

$C = - \frac{3}{4}$

$B = \frac{11}{4}$

$A = - 1$

$C = - \frac{3}{4}$

$B = \frac{11}{4}$

$A = - 1$

Step 1.3.6

Solve the system of equations.

$C = - \frac{3}{4} B = \frac{11}{4} A = - 1$

Step 1.3.7

List all of the solutions.

$C = - \frac{3}{4}, B = \frac{11}{4}, A = - 1$

Step 1.4

Replace each of the partial fraction coefficients in $\frac{A}{x^{2}} + \frac{B}{x} + \frac{C}{x - 4}$ with the values found for $A$ , $B$ , and $C$ .

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

Step 1.5

Simplify.

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Step 1.5.1

Move the negative in front of the fraction.

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

Step 1.5.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{x^{2}} + \frac{11}{4} \cdot \frac{1}{x} + \frac{- \frac{3}{4}}{x - 4}$

Step 1.5.3

Multiply $\frac{11}{4}$ by $\frac{1}{x}$ .

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

Step 1.5.4

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{x^{2}} + \frac{11}{4 x} - \frac{3}{4} \cdot \frac{1}{x - 4}$

Step 1.5.5

Multiply $\frac{1}{x - 4}$ by $\frac{3}{4}$ .

$- \frac{1}{x^{2}} + \frac{11}{4 x} - \frac{3}{(x - 4) \cdot 4}$

Step 1.5.6

Move $4$ to the left of $x - 4$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \int x^{2 \cdot - 1} d x + \int \frac{11}{4 x} d x + \int - \frac{3}{4 (x - 4)} d x$

Step 4.2.2

Multiply $2$ by $- 1$ .

$- \int x^{- 2} d x + \int \frac{11}{4 x} d x + \int - \frac{3}{4 (x - 4)} d x$

Step 5

By the Power Rule, the integral of $x^{- 2}$ with respect to $x$ is $- x^{- 1}$ .

$- (- x^{- 1} + C) + \int \frac{11}{4 x} d x + \int - \frac{3}{4 (x - 4)} d x$

Step 6

Since $\frac{11}{4}$ is constant with respect to $x$ , move $\frac{11}{4}$ out of the integral.

$- (- x^{- 1} + C) + \frac{11}{4} \int \frac{1}{x} d x + \int - \frac{3}{4 (x - 4)} d x$

Step 7

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

$- (- x^{- 1} + C) + \frac{11}{4} (\ln (| x |) + C) + \int - \frac{3}{4 (x - 4)} d x$

Step 8

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

$- (- x^{- 1} + C) + \frac{11}{4} (\ln (| x |) + C) - \int \frac{3}{4 (x - 4)} d x$

Step 9

Since $\frac{3}{4}$ is constant with respect to $x$ , move $\frac{3}{4}$ out of the integral.

$- (- x^{- 1} + C) + \frac{11}{4} (\ln (| x |) + C) - (\frac{3}{4} \int \frac{1}{x - 4} d x)$

Step 10

Let $u = x - 4$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 10.1

Let $u = x - 4$ . Find $\frac{d u}{d x}$ .

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Step 10.1.1

Differentiate $x - 4$ .

$\frac{d}{d x} [x - 4]$

Step 10.1.2

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$

Step 10.1.3

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

$1 + \frac{d}{d x} [- 4]$

Step 10.1.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$1 + 0$

Step 10.1.5

Add $1$ and $0$ .

$1$

Step 10.2

Rewrite the problem using $u$ and $d u$ .

$- (- x^{- 1} + C) + \frac{11}{4} (\ln (| x |) + C) - \frac{3}{4} \int \frac{1}{u} d u$

Step 11

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

$- (- x^{- 1} + C) + \frac{11}{4} (\ln (| x |) + C) - \frac{3}{4} (\ln (| u |) + C)$

Step 12

Simplify.

$\frac{1}{x} + \frac{11 \ln (| x |)}{4} - \frac{3}{4} \ln (| u |) + C$

Step 13

Replace all occurrences of $u$ with $x - 4$ .

$\frac{1}{x} + \frac{11 \ln (| x |)}{4} - \frac{3}{4} \ln (| x - 4 |) + C$

Step 14

Reorder terms.

$\frac{1}{x} + \frac{11}{4} \ln (| x |) - \frac{3}{4} \ln (| x - 4 |) + C$