Calculus Examples

$\int \frac{x^{2} - x + 9}{{(x^{2} + 9)}^{2}} d x$

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

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.

$\frac{A x + B}{{(x^{2} + 9)}^{2}}$

Step 1.1.2

$\frac{A x + B}{{(x^{2} + 9)}^{2}} + \frac{C x + D}{x^{2} + 9}$

Step 1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x^{2} + 9)}^{2}$ .

$\frac{(x^{2} - x + 9) {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{2}} = \frac{(A x + B) {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{2}} + \frac{(C x + D) {(x^{2} + 9)}^{2}}{x^{2} + 9}$

Step 1.1.4

Cancel the common factor of ${(x^{2} + 9)}^{2}$ .

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Step 1.1.4.1

Cancel the common factor.

$\frac{(x^{2} - x + 9) {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{2}} = \frac{(A x + B) {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{2}} + \frac{(C x + D) {(x^{2} + 9)}^{2}}{x^{2} + 9}$

Step 1.1.4.2

Divide $x^{2} - x + 9$ by $1$ .

$x^{2} - x + 9 = \frac{(A x + B) {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{2}} + \frac{(C x + D) {(x^{2} + 9)}^{2}}{x^{2} + 9}$

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

Cancel the common factor of ${(x^{2} + 9)}^{2}$ .

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Step 1.1.5.1.1

Cancel the common factor.

$x^{2} - x + 9 = \frac{(A x + B) {(x^{2} + 9)}^{2}}{{(x^{2} + 9)}^{2}} + \frac{(C x + D) {(x^{2} + 9)}^{2}}{x^{2} + 9}$

Step 1.1.5.1.2

Divide $A x + B$ by $1$ .

$x^{2} - x + 9 = A x + B + \frac{(C x + D) {(x^{2} + 9)}^{2}}{x^{2} + 9}$

Step 1.1.5.2

Cancel the common factor of ${(x^{2} + 9)}^{2}$ and $x^{2} + 9$ .

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Step 1.1.5.2.1

Factor $x^{2} + 9$ out of $(C x + D) {(x^{2} + 9)}^{2}$ .

$x^{2} - x + 9 = A x + B + \frac{(x^{2} + 9) ((C x + D) (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.5.2.2

Cancel the common factors.

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Step 1.1.5.2.2.1

Multiply by $1$ .

$x^{2} - x + 9 = A x + B + \frac{(x^{2} + 9) ((C x + D) (x^{2} + 9))}{(x^{2} + 9) \cdot 1}$

Step 1.1.5.2.2.2

Cancel the common factor.

$x^{2} - x + 9 = A x + B + \frac{(x^{2} + 9) ((C x + D) (x^{2} + 9))}{(x^{2} + 9) \cdot 1}$

Step 1.1.5.2.2.3

Rewrite the expression.

$x^{2} - x + 9 = A x + B + \frac{(C x + D) (x^{2} + 9)}{1}$

Step 1.1.5.2.2.4

Divide $(C x + D) (x^{2} + 9)$ by $1$ .

$x^{2} - x + 9 = A x + B + (C x + D) (x^{2} + 9)$

Step 1.1.5.3

Expand $(C x + D) (x^{2} + 9)$ using the FOIL Method.

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Step 1.1.5.3.1

Apply the distributive property.

$x^{2} - x + 9 = A x + B + C x (x^{2} + 9) + D (x^{2} + 9)$

Step 1.1.5.3.2

Apply the distributive property.

$x^{2} - x + 9 = A x + B + C x \cdot x^{2} + C x \cdot 9 + D (x^{2} + 9)$

Step 1.1.5.3.3

Apply the distributive property.

$x^{2} - x + 9 = A x + B + C x \cdot x^{2} + C x \cdot 9 + D x^{2} + D \cdot 9$

Step 1.1.5.4

Simplify each term.

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Step 1.1.5.4.1

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

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Step 1.1.5.4.1.1

Move $x^{2}$ .

$x^{2} - x + 9 = A x + B + C (x^{2} x) + C x \cdot 9 + D x^{2} + D \cdot 9$

Step 1.1.5.4.1.2

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

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Step 1.1.5.4.1.2.1

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

$x^{2} - x + 9 = A x + B + C (x^{2} x) + C x \cdot 9 + D x^{2} + D \cdot 9$

Step 1.1.5.4.1.2.2

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

$x^{2} - x + 9 = A x + B + C x^{2 + 1} + C x \cdot 9 + D x^{2} + D \cdot 9$

Step 1.1.5.4.1.3

Add $2$ and $1$ .

$x^{2} - x + 9 = A x + B + C x^{3} + C x \cdot 9 + D x^{2} + D \cdot 9$

Step 1.1.5.4.2

Move $9$ to the left of $C x$ .

$x^{2} - x + 9 = A x + B + C x^{3} + 9 \cdot (C x) + D x^{2} + D \cdot 9$

Step 1.1.5.4.3

Move $9$ to the left of $D$ .

$x^{2} - x + 9 = A x + B + C x^{3} + 9 C x + D x^{2} + 9 D$

Step 1.1.6

Simplify the expression.

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Step 1.1.6.1

Move $B$ .

$x^{2} - x + 9 = A x + C x^{3} + 9 C x + D x^{2} + B + 9 D$

Step 1.1.6.2

Move $9 C x$ .

$x^{2} - x + 9 = A x + C x^{3} + D x^{2} + 9 C x + B + 9 D$

Step 1.1.6.3

Move $A x$ .

$x^{2} - x + 9 = C x^{3} + D x^{2} + A x + 9 C x + B + 9 D$

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.

$0 = C$

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.

$1 = D$

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.

$- 1 = A + 9 C$

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.

$9 = B + 9 D$

Step 1.2.5

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

$0 = C$

$1 = D$

$- 1 = A + 9 C$

$9 = B + 9 D$

$0 = C$

$1 = D$

$- 1 = A + 9 C$

$9 = B + 9 D$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$1 = D$

$- 1 = A + 9 C$

$9 = B + 9 D$

Step 1.3.2

Replace all occurrences of $C$ with $0$ in each equation.

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Step 1.3.2.1

Rewrite the equation as $D = 1$ .

$D = 1$

$C = 0$

$- 1 = A + 9 C$

$9 = B + 9 D$

Step 1.3.2.2

Replace all occurrences of $C$ in $- 1 = A + 9 C$ with $0$ .

$- 1 = A + 9 (0)$

$D = 1$

$C = 0$

$9 = B + 9 D$

Step 1.3.2.3

Simplify the right side.

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Step 1.3.2.3.1

Simplify $A + 9 (0)$ .

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Step 1.3.2.3.1.1

Multiply $9$ by $0$ .

$- 1 = A + 0$

$D = 1$

$C = 0$

$9 = B + 9 D$

Step 1.3.2.3.1.2

Add $A$ and $0$ .

$- 1 = A$

$D = 1$

$C = 0$

$9 = B + 9 D$

$- 1 = A$

$D = 1$

$C = 0$

$9 = B + 9 D$

$- 1 = A$

$D = 1$

$C = 0$

$9 = B + 9 D$

$- 1 = A$

$D = 1$

$C = 0$

$9 = B + 9 D$

Step 1.3.3

Replace all occurrences of $D$ with $1$ in each equation.

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Step 1.3.3.1

Rewrite the equation as $A = - 1$ .

$A = - 1$

$D = 1$

$C = 0$

$9 = B + 9 D$

Step 1.3.3.2

Replace all occurrences of $D$ in $9 = B + 9 D$ with $1$ .

$9 = B + 9 (1)$

$A = - 1$

$D = 1$

$C = 0$

Step 1.3.3.3

Simplify the right side.

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Step 1.3.3.3.1

Multiply $9$ by $1$ .

$9 = B + 9$

$A = - 1$

$D = 1$

$C = 0$

$9 = B + 9$

$A = - 1$

$D = 1$

$C = 0$

$9 = B + 9$

$A = - 1$

$D = 1$

$C = 0$

Step 1.3.4

Solve for $B$ in $9 = B + 9$ .

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Step 1.3.4.1

Rewrite the equation as $B + 9 = 9$ .

$B + 9 = 9$

$A = - 1$

$D = 1$

$C = 0$

Step 1.3.4.2

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

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Step 1.3.4.2.1

Subtract $9$ from both sides of the equation.

$B = 9 - 9$

$A = - 1$

$D = 1$

$C = 0$

Step 1.3.4.2.2

Subtract $9$ from $9$ .

$B = 0$

$A = - 1$

$D = 1$

$C = 0$

$B = 0$

$A = - 1$

$D = 1$

$C = 0$

$B = 0$

$A = - 1$

$D = 1$

$C = 0$

Step 1.3.5

Solve the system of equations.

$B = 0 A = - 1 D = 1 C = 0$

Step 1.3.6

List all of the solutions.

$B = 0, A = - 1, D = 1, C = 0$

Step 1.4

Replace each of the partial fraction coefficients in $\frac{A x + B}{{(x^{2} + 9)}^{2}} + \frac{C x + D}{x^{2} + 9}$ with the values found for $A$ , $B$ , $C$ , and $D$ .

$\frac{- 1 x + 0}{{(x^{2} + 9)}^{2}} + \frac{0 x + 1}{x^{2} + 9}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Rewrite $- 1 x$ as $- x$ .

$\frac{- x + 0}{{(x^{2} + 9)}^{2}} + \frac{(0) \cdot x + 1}{x^{2} + 9}$

Step 1.5.1.2

Add $- x$ and $0$ .

$\frac{- x}{{(x^{2} + 9)}^{2}} + \frac{(0) \cdot x + 1}{x^{2} + 9}$

Step 1.5.2

Move the negative in front of the fraction.

$- \frac{x}{{(x^{2} + 9)}^{2}} + \frac{(0) \cdot x + 1}{x^{2} + 9}$

Step 1.5.3

Simplify the numerator.

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Step 1.5.3.1

Multiply $0$ by $x$ .

$- \frac{x}{{(x^{2} + 9)}^{2}} + \frac{0 + 1}{x^{2} + 9}$

Step 1.5.3.2

Add $0$ and $1$ .

$\int - \frac{x}{{(x^{2} + 9)}^{2}} + \frac{1}{x^{2} + 9} d x$

Step 2

Split the single integral into multiple integrals.

$\int - \frac{x}{{(x^{2} + 9)}^{2}} d x + \int \frac{1}{x^{2} + 9} d x$

Step 3

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

$- \int \frac{x}{{(x^{2} + 9)}^{2}} d x + \int \frac{1}{x^{2} + 9} d x$

Step 4

Let $u = x^{2} + 9$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x^{2} + 9$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x^{2} + 9$ .

$\frac{d}{d x} [x^{2} + 9]$

Step 4.1.2

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [9]$

Step 4.1.4

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$2 x + 0$

Step 4.1.5

Add $2 x$ and $0$ .

$2 x$

Step 4.2

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

$- \int \frac{1}{u^{2}} \cdot \frac{1}{2} d u + \int \frac{1}{x^{2} + 9} d x$

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{2}$ .

$- \int \frac{1}{u^{2} \cdot 2} d u + \int \frac{1}{x^{2} + 9} d x$

Step 5.2

Move $2$ to the left of $u^{2}$ .

$- \int \frac{1}{2 u^{2}} d u + \int \frac{1}{x^{2} + 9} d x$

Step 6

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int \frac{1}{u^{2}} d u) + \int \frac{1}{x^{2} + 9} d x$

Step 7

Apply basic rules of exponents.

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Step 7.1

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

$- \frac{1}{2} \int {(u^{2})}^{- 1} d u + \int \frac{1}{x^{2} + 9} d x$

Step 7.2

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

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Step 7.2.1

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

$- \frac{1}{2} \int u^{2 \cdot - 1} d u + \int \frac{1}{x^{2} + 9} d x$

Step 7.2.2

Multiply $2$ by $- 1$ .

$- \frac{1}{2} \int u^{- 2} d u + \int \frac{1}{x^{2} + 9} d x$

Step 8

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

$- \frac{1}{2} (- u^{- 1} + C) + \int \frac{1}{x^{2} + 9} d x$

Step 9

Simplify the expression.

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Step 9.1

Reorder $x^{2}$ and $9$ .

$- \frac{1}{2} (- u^{- 1} + C) + \int \frac{1}{9 + x^{2}} d x$

Step 9.2

Rewrite $9$ as $3^{2}$ .

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

Step 10

The integral of $\frac{1}{3^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{3} \arctan (\frac{x}{3}) + C$ .

$- \frac{1}{2} (- u^{- 1} + C) + \frac{1}{3} \arctan (\frac{x}{3}) + C$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{3}$ and $\arctan (\frac{x}{3})$ .

$- \frac{1}{2} (- u^{- 1} + C) + \frac{\arctan (\frac{x}{3})}{3} + C$

Step 11.2

Simplify.

$- \frac{1}{2} (- \frac{1}{u}) + \frac{\arctan (\frac{x}{3})}{3} + C$

Step 11.3

Simplify.

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Step 11.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) \frac{1}{u} + \frac{\arctan (\frac{x}{3})}{3} + C$

Step 11.3.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} \cdot \frac{1}{u} + \frac{\arctan (\frac{x}{3})}{3} + C$

Step 11.3.3

Multiply $\frac{1}{2}$ by $\frac{1}{u}$ .

$\frac{1}{2 u} + \frac{\arctan (\frac{x}{3})}{3} + C$

Step 12

Replace all occurrences of $u$ with $x^{2} + 9$ .

$\frac{1}{2 (x^{2} + 9)} + \frac{\arctan (\frac{x}{3})}{3} + C$

Step 13

Reorder terms.

$\frac{1}{2 (x^{2} + 9)} + \frac{1}{3} \arctan (\frac{1}{3} x) + C$