Calculus Examples

$\int \frac{7 x^{2} - 9 x + 54}{(x - 3) \cdot (x^{2} + 9)} d x$

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

$\frac{A}{x - 3}$

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.

$\frac{A}{x - 3} + \frac{B x + C}{x^{2} + 9}$

Step 1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(x - 3) \cdot (x^{2} + 9)$ .

$\frac{(7 x^{2} - 9 x + 54) ((x - 3) \cdot (x^{2} + 9))}{(x - 3) \cdot (x^{2} + 9)} = \frac{(A) ((x - 3) \cdot (x^{2} + 9))}{x - 3} + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.4

Cancel the common factor of $x - 3$ .

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Step 1.1.4.1

Cancel the common factor.

$\frac{(7 x^{2} - 9 x + 54) ((x - 3) \cdot (x^{2} + 9))}{(x - 3) \cdot (x^{2} + 9)} = \frac{(A) ((x - 3) \cdot (x^{2} + 9))}{x - 3} + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.4.2

Rewrite the expression.

$\frac{(7 x^{2} - 9 x + 54) (x^{2} + 9)}{x^{2} + 9} = \frac{(A) ((x - 3) \cdot (x^{2} + 9))}{x - 3} + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.5

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

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Step 1.1.5.1

Cancel the common factor.

$\frac{(7 x^{2} - 9 x + 54) (x^{2} + 9)}{x^{2} + 9} = \frac{(A) ((x - 3) \cdot (x^{2} + 9))}{x - 3} + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.5.2

Divide $7 x^{2} - 9 x + 54$ by $1$ .

$7 x^{2} - 9 x + 54 = \frac{(A) ((x - 3) \cdot (x^{2} + 9))}{x - 3} + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.6

Simplify each term.

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Step 1.1.6.1

Cancel the common factor of $x - 3$ .

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Step 1.1.6.1.1

Cancel the common factor.

$7 x^{2} - 9 x + 54 = \frac{A ((x - 3) \cdot (x^{2} + 9))}{x - 3} + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.6.1.2

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

$7 x^{2} - 9 x + 54 = (A) (x^{2} + 9) + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.6.2

Apply the distributive property.

$7 x^{2} - 9 x + 54 = A x^{2} + A \cdot 9 + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.6.3

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

$7 x^{2} - 9 x + 54 = A x^{2} + 9 \cdot A + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.6.4

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

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Step 1.1.6.4.1

Cancel the common factor.

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + \frac{(B x + C) ((x - 3) \cdot (x^{2} + 9))}{x^{2} + 9}$

Step 1.1.6.4.2

Divide $(B x + C) (x - 3)$ by $1$ .

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + (B x + C) (x - 3)$

Step 1.1.6.5

Expand $(B x + C) (x - 3)$ using the FOIL Method.

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Step 1.1.6.5.1

Apply the distributive property.

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + B x (x - 3) + C (x - 3)$

Step 1.1.6.5.2

Apply the distributive property.

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + B x \cdot x + B x \cdot - 3 + C (x - 3)$

Step 1.1.6.5.3

Apply the distributive property.

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + B x \cdot x + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.1.6.6

Simplify each term.

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Step 1.1.6.6.1

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

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Step 1.1.6.6.1.1

Move $x$ .

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + B (x \cdot x) + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.1.6.6.1.2

Multiply $x$ by $x$ .

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + B x^{2} + B x \cdot - 3 + C x + C \cdot - 3$

Step 1.1.6.6.2

Move $- 3$ to the left of $B x$ .

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + B x^{2} - 3 \cdot (B x) + C x + C \cdot - 3$

Step 1.1.6.6.3

Move $- 3$ to the left of $C$ .

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + B x^{2} - 3 B x + C x - 3 C$

Step 1.1.7

Simplify the expression.

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Step 1.1.7.1

Move $B$ .

$7 x^{2} - 9 x + 54 = A x^{2} + 9 A + B x^{2} - 3 x B + C x - 3 C$

Step 1.1.7.2

Move $9 A$ .

$7 x^{2} - 9 x + 54 = A x^{2} + B x^{2} - 3 x B + C x + 9 A - 3 C$

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.

$7 = A + B$

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.

$- 9 = - 3 B + C$

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.

$54 = 9 A - 3 C$

Step 1.2.4

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

$7 = A + B$

$- 9 = - 3 B + C$

$54 = 9 A - 3 C$

$7 = A + B$

$- 9 = - 3 B + C$

$54 = 9 A - 3 C$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Solve for $A$ in $7 = A + B$ .

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Step 1.3.1.1

Rewrite the equation as $A + B = 7$ .

$A + B = 7$

$- 9 = - 3 B + C$

$54 = 9 A - 3 C$

Step 1.3.1.2

Subtract $B$ from both sides of the equation.

$A = 7 - B$

$- 9 = - 3 B + C$

$54 = 9 A - 3 C$

$A = 7 - B$

$- 9 = - 3 B + C$

$54 = 9 A - 3 C$

Step 1.3.2

Replace all occurrences of $A$ with $7 - B$ in each equation.

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Step 1.3.2.1

Replace all occurrences of $A$ in $54 = 9 A - 3 C$ with $7 - B$ .

$54 = 9 (7 - B) - 3 C$

$A = 7 - B$

$- 9 = - 3 B + C$

Step 1.3.2.2

Simplify the right side.

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Step 1.3.2.2.1

Simplify each term.

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Step 1.3.2.2.1.1

Apply the distributive property.

$54 = 9 \cdot 7 + 9 (- B) - 3 C$

$A = 7 - B$

$- 9 = - 3 B + C$

Step 1.3.2.2.1.2

Multiply $9$ by $7$ .

$54 = 63 + 9 (- B) - 3 C$

$A = 7 - B$

$- 9 = - 3 B + C$

Step 1.3.2.2.1.3

Multiply $- 1$ by $9$ .

$54 = 63 - 9 B - 3 C$

$A = 7 - B$

$- 9 = - 3 B + C$

$54 = 63 - 9 B - 3 C$

$A = 7 - B$

$- 9 = - 3 B + C$

$54 = 63 - 9 B - 3 C$

$A = 7 - B$

$- 9 = - 3 B + C$

$54 = 63 - 9 B - 3 C$

$A = 7 - B$

$- 9 = - 3 B + C$

Step 1.3.3

Reorder $7$ and $- B$ .

$A = - B + 7$

$54 = 63 - 9 B - 3 C$

$- 9 = - 3 B + C$

Step 1.3.4

Solve for $C$ in $- 9 = - 3 B + C$ .

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Step 1.3.4.1

Rewrite the equation as $- 3 B + C = - 9$ .

$- 3 B + C = - 9$

$A = - B + 7$

$54 = 63 - 9 B - 3 C$

Step 1.3.4.2

Add $3 B$ to both sides of the equation.

$C = - 9 + 3 B$

$A = - B + 7$

$54 = 63 - 9 B - 3 C$

$C = - 9 + 3 B$

$A = - B + 7$

$54 = 63 - 9 B - 3 C$

Step 1.3.5

Replace all occurrences of $C$ with $- 9 + 3 B$ in each equation.

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Step 1.3.5.1

Replace all occurrences of $C$ in $54 = 63 - 9 B - 3 C$ with $- 9 + 3 B$ .

$54 = 63 - 9 B - 3 (- 9 + 3 B)$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.5.2

Simplify the right side.

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Step 1.3.5.2.1

Simplify $63 - 9 B - 3 (- 9 + 3 B)$ .

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Step 1.3.5.2.1.1

Simplify each term.

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Step 1.3.5.2.1.1.1

Apply the distributive property.

$54 = 63 - 9 B - 3 \cdot - 9 - 3 (3 B)$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.5.2.1.1.2

Multiply $- 3$ by $- 9$ .

$54 = 63 - 9 B + 27 - 3 (3 B)$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.5.2.1.1.3

Multiply $3$ by $- 3$ .

$54 = 63 - 9 B + 27 - 9 B$

$C = - 9 + 3 B$

$A = - B + 7$

$54 = 63 - 9 B + 27 - 9 B$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.5.2.1.2

Simplify by adding terms.

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Step 1.3.5.2.1.2.1

Add $63$ and $27$ .

$54 = - 9 B + 90 - 9 B$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.5.2.1.2.2

Subtract $9 B$ from $- 9 B$ .

$54 = - 18 B + 90$

$C = - 9 + 3 B$

$A = - B + 7$

$54 = - 18 B + 90$

$C = - 9 + 3 B$

$A = - B + 7$

$54 = - 18 B + 90$

$C = - 9 + 3 B$

$A = - B + 7$

$54 = - 18 B + 90$

$C = - 9 + 3 B$

$A = - B + 7$

$54 = - 18 B + 90$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.6

Solve for $B$ in $54 = - 18 B + 90$ .

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Step 1.3.6.1

Rewrite the equation as $- 18 B + 90 = 54$ .

$- 18 B + 90 = 54$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.6.2

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

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Step 1.3.6.2.1

Subtract $90$ from both sides of the equation.

$- 18 B = 54 - 90$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.6.2.2

Subtract $90$ from $54$ .

$- 18 B = - 36$

$C = - 9 + 3 B$

$A = - B + 7$

$- 18 B = - 36$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.6.3

Divide each term in $- 18 B = - 36$ by $- 18$ and simplify.

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Step 1.3.6.3.1

Divide each term in $- 18 B = - 36$ by $- 18$ .

$\frac{- 18 B}{- 18} = \frac{- 36}{- 18}$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.6.3.2

Simplify the left side.

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Step 1.3.6.3.2.1

Cancel the common factor of $- 18$ .

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Step 1.3.6.3.2.1.1

Cancel the common factor.

$\frac{- 18 B}{- 18} = \frac{- 36}{- 18}$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.6.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{- 36}{- 18}$

$C = - 9 + 3 B$

$A = - B + 7$

$B = \frac{- 36}{- 18}$

$C = - 9 + 3 B$

$A = - B + 7$

$B = \frac{- 36}{- 18}$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.6.3.3

Simplify the right side.

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Step 1.3.6.3.3.1

Divide $- 36$ by $- 18$ .

$B = 2$

$C = - 9 + 3 B$

$A = - B + 7$

$B = 2$

$C = - 9 + 3 B$

$A = - B + 7$

$B = 2$

$C = - 9 + 3 B$

$A = - B + 7$

$B = 2$

$C = - 9 + 3 B$

$A = - B + 7$

Step 1.3.7

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

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Step 1.3.7.1

Replace all occurrences of $B$ in $C = - 9 + 3 B$ with $2$ .

$C = - 9 + 3 (2)$

$B = 2$

$A = - B + 7$

Step 1.3.7.2

Simplify the right side.

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Step 1.3.7.2.1

Simplify $- 9 + 3 (2)$ .

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Step 1.3.7.2.1.1

Multiply $3$ by $2$ .

$C = - 9 + 6$

$B = 2$

$A = - B + 7$

Step 1.3.7.2.1.2

Add $- 9$ and $6$ .

$C = - 3$

$B = 2$

$A = - B + 7$

$C = - 3$

$B = 2$

$A = - B + 7$

$C = - 3$

$B = 2$

$A = - B + 7$

Step 1.3.7.3

Replace all occurrences of $B$ in $A = - B + 7$ with $2$ .

$A = - (2) + 7$

$C = - 3$

$B = 2$

Step 1.3.7.4

Simplify the right side.

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Step 1.3.7.4.1

Simplify $- (2) + 7$ .

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Step 1.3.7.4.1.1

Multiply $- 1$ by $2$ .

$A = - 2 + 7$

$C = - 3$

$B = 2$

Step 1.3.7.4.1.2

Add $- 2$ and $7$ .

$A = 5$

$C = - 3$

$B = 2$

$A = 5$

$C = - 3$

$B = 2$

$A = 5$

$C = - 3$

$B = 2$

$A = 5$

$C = - 3$

$B = 2$

Step 1.3.8

List all of the solutions.

$A = 5, C = - 3, B = 2$

Step 1.4

Replace each of the partial fraction coefficients in $\frac{A}{x - 3} + \frac{B x + C}{x^{2} + 9}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{5}{x - 3} + \frac{2 x - 3}{x^{2} + 9}$

Step 1.5

Multiply $2$ by $x$ .

$\int \frac{5}{x - 3} + \frac{2 x - 3}{x^{2} + 9} d x$

Step 2

Split the single integral into multiple integrals.

$\int \frac{5}{x - 3} d x + \int \frac{2 x - 3}{x^{2} + 9} d x$

Step 3

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int \frac{1}{x - 3} d x + \int \frac{2 x - 3}{x^{2} + 9} d x$

Step 4

Let $u_{1} = x - 3$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.1

Let $u_{1} = x - 3$ . Find $\frac{d u_{1}}{d x}$ .

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Step 4.1.1

Differentiate $x - 3$ .

$\frac{d}{d x} [x - 3]$

Step 4.1.2

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Step 4.1.3

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

$1 + \frac{d}{d x} [- 3]$

Step 4.1.4

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

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

$5 \int \frac{1}{u_{1}} d u_{1} + \int \frac{2 x - 3}{x^{2} + 9} d x$

Step 5

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

$5 (\ln (| u_{1} |) + C) + \int \frac{2 x - 3}{x^{2} + 9} d x$

Step 6

Split the fraction $\frac{2 x - 3}{x^{2} + 9}$ into two fractions.

$5 (\ln (| u_{1} |) + C) + \int \frac{2 x}{x^{2} + 9} + \frac{- 3}{x^{2} + 9} d x$

Step 7

Move the negative in front of the fraction.

$5 (\ln (| u_{1} |) + C) + \int \frac{2 x}{x^{2} + 9} - \frac{3}{x^{2} + 9} d x$

Step 8

Split the single integral into multiple integrals.

$5 (\ln (| u_{1} |) + C) + \int \frac{2 x}{x^{2} + 9} d x + \int - \frac{3}{x^{2} + 9} d x$

Step 9

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

$5 (\ln (| u_{1} |) + C) + 2 \int \frac{x}{x^{2} + 9} d x + \int - \frac{3}{x^{2} + 9} d x$

Step 10

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

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Step 10.1

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

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Step 10.1.1

Differentiate $x^{2} + 9$ .

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

Step 10.1.2

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 10.1.3

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 10.1.4

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

$2 x + 0$

Step 10.1.5

Add $2 x$ and $0$ .

$2 x$

Step 10.2

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

$5 (\ln (| u_{1} |) + C) + 2 \int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2} + \int - \frac{3}{x^{2} + 9} d x$

Step 11

Simplify.

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Step 11.1

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

$5 (\ln (| u_{1} |) + C) + 2 \int \frac{1}{u_{2} \cdot 2} d u_{2} + \int - \frac{3}{x^{2} + 9} d x$

Step 11.2

Move $2$ to the left of $u_{2}$ .

$5 (\ln (| u_{1} |) + C) + 2 \int \frac{1}{2 u_{2}} d u_{2} + \int - \frac{3}{x^{2} + 9} d x$

Step 12

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

$5 (\ln (| u_{1} |) + C) + 2 (\frac{1}{2} \int \frac{1}{u_{2}} d u_{2}) + \int - \frac{3}{x^{2} + 9} d x$

Step 13

Simplify.

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Step 13.1

Combine $\frac{1}{2}$ and $2$ .

$5 (\ln (| u_{1} |) + C) + \frac{2}{2} \int \frac{1}{u_{2}} d u_{2} + \int - \frac{3}{x^{2} + 9} d x$

Step 13.2

Cancel the common factor of $2$ .

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Step 13.2.1

Cancel the common factor.

$5 (\ln (| u_{1} |) + C) + \frac{2}{2} \int \frac{1}{u_{2}} d u_{2} + \int - \frac{3}{x^{2} + 9} d x$

Step 13.2.2

Rewrite the expression.

$5 (\ln (| u_{1} |) + C) + 1 \int \frac{1}{u_{2}} d u_{2} + \int - \frac{3}{x^{2} + 9} d x$

Step 13.3

Multiply $\int \frac{1}{u_{2}} d u_{2}$ by $1$ .

$5 (\ln (| u_{1} |) + C) + \int \frac{1}{u_{2}} d u_{2} + \int - \frac{3}{x^{2} + 9} d x$

Step 14

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

$5 (\ln (| u_{1} |) + C) + \ln (| u_{2} |) + C + \int - \frac{3}{x^{2} + 9} d x$

Step 15

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

$5 (\ln (| u_{1} |) + C) + \ln (| u_{2} |) + C - \int \frac{3}{x^{2} + 9} d x$

Step 16

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$5 (\ln (| u_{1} |) + C) + \ln (| u_{2} |) + C - (3 \int \frac{1}{x^{2} + 9} d x)$

Step 17

Simplify the expression.

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Step 17.1

Multiply $3$ by $- 1$ .

$5 (\ln (| u_{1} |) + C) + \ln (| u_{2} |) + C - 3 \int \frac{1}{x^{2} + 9} d x$

Step 17.2

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

$5 (\ln (| u_{1} |) + C) + \ln (| u_{2} |) + C - 3 \int \frac{1}{9 + x^{2}} d x$

Step 17.3

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

$5 (\ln (| u_{1} |) + C) + \ln (| u_{2} |) + C - 3 \int \frac{1}{3^{2} + x^{2}} d x$

Step 18

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

$5 (\ln (| u_{1} |) + C) + \ln (| u_{2} |) + C - 3 (\frac{1}{3} \arctan (\frac{x}{3}) + C)$

Step 19

Simplify.

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Step 19.1

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

$5 (\ln (| u_{1} |) + C) + \ln (| u_{2} |) + C - 3 (\frac{\arctan (\frac{x}{3})}{3} + C)$

Step 19.2

Simplify.

$5 \ln (| u_{1} |) + \ln (| u_{2} |) - \arctan (\frac{x}{3}) + C$

Step 20

Substitute back in for each integration substitution variable.

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Step 20.1

Replace all occurrences of $u_{1}$ with $x - 3$ .

$5 \ln (| x - 3 |) + \ln (| u_{2} |) - \arctan (\frac{x}{3}) + C$

Step 20.2

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

$5 \ln (| x - 3 |) + \ln (| x^{2} + 9 |) - \arctan (\frac{x}{3}) + C$

Step 21

Reorder terms.

$5 \ln (| x - 3 |) + \ln (| x^{2} + 9 |) - \arctan (\frac{1}{3} x) + C$