Calculus Examples

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

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 $A$ .

$\frac{A}{2 x + 3}$

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.

$\frac{A}{2 x + 3} + \frac{B x + C}{x^{2} + 4}$

Step 1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $(2 x + 3) (x^{2} + 4)$ .

$\frac{(2 x^{2} + 2 x + 11) (2 x + 3) (x^{2} + 4)}{(2 x + 3) (x^{2} + 4)} = \frac{(A) (2 x + 3) (x^{2} + 4)}{2 x + 3} + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.4

Cancel the common factor of $2 x + 3$ .

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Step 1.1.4.1

Cancel the common factor.

$\frac{(2 x^{2} + 2 x + 11) (2 x + 3) (x^{2} + 4)}{(2 x + 3) (x^{2} + 4)} = \frac{(A) (2 x + 3) (x^{2} + 4)}{2 x + 3} + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.4.2

Rewrite the expression.

$\frac{(2 x^{2} + 2 x + 11) (x^{2} + 4)}{x^{2} + 4} = \frac{(A) (2 x + 3) (x^{2} + 4)}{2 x + 3} + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.5

Cancel the common factor of $x^{2} + 4$ .

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Step 1.1.5.1

Cancel the common factor.

$\frac{(2 x^{2} + 2 x + 11) (x^{2} + 4)}{x^{2} + 4} = \frac{(A) (2 x + 3) (x^{2} + 4)}{2 x + 3} + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.5.2

Divide $2 x^{2} + 2 x + 11$ by $1$ .

$2 x^{2} + 2 x + 11 = \frac{(A) (2 x + 3) (x^{2} + 4)}{2 x + 3} + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.6

Simplify each term.

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Step 1.1.6.1

Cancel the common factor of $2 x + 3$ .

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Step 1.1.6.1.1

Cancel the common factor.

$2 x^{2} + 2 x + 11 = \frac{A (2 x + 3) (x^{2} + 4)}{2 x + 3} + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.6.1.2

Divide $(A) (x^{2} + 4)$ by $1$ .

$2 x^{2} + 2 x + 11 = (A) (x^{2} + 4) + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.6.2

Apply the distributive property.

$2 x^{2} + 2 x + 11 = A x^{2} + A \cdot 4 + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.6.3

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

$2 x^{2} + 2 x + 11 = A x^{2} + 4 \cdot A + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.6.4

Cancel the common factor of $x^{2} + 4$ .

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Step 1.1.6.4.1

Cancel the common factor.

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + \frac{(B x + C) (2 x + 3) (x^{2} + 4)}{x^{2} + 4}$

Step 1.1.6.4.2

Divide $(B x + C) (2 x + 3)$ by $1$ .

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + (B x + C) (2 x + 3)$

Step 1.1.6.5

Expand $(B x + C) (2 x + 3)$ using the FOIL Method.

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Step 1.1.6.5.1

Apply the distributive property.

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + B x (2 x + 3) + C (2 x + 3)$

Step 1.1.6.5.2

Apply the distributive property.

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + B x (2 x) + B x \cdot 3 + C (2 x + 3)$

Step 1.1.6.5.3

Apply the distributive property.

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + B x (2 x) + B x \cdot 3 + C (2 x) + C \cdot 3$

Step 1.1.6.6

Simplify each term.

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Step 1.1.6.6.1

Rewrite using the commutative property of multiplication.

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + 2 B x \cdot x + B x \cdot 3 + C (2 x) + C \cdot 3$

Step 1.1.6.6.2

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

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Step 1.1.6.6.2.1

Move $x$ .

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + 2 B (x \cdot x) + B x \cdot 3 + C (2 x) + C \cdot 3$

Step 1.1.6.6.2.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + 2 B x^{2} + B x \cdot 3 + C (2 x) + C \cdot 3$

Step 1.1.6.6.3

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

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + 2 B x^{2} + 3 \cdot (B x) + C (2 x) + C \cdot 3$

Step 1.1.6.6.4

Rewrite using the commutative property of multiplication.

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + 2 B x^{2} + 3 B x + 2 C x + C \cdot 3$

Step 1.1.6.6.5

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

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + 2 B x^{2} + 3 B x + 2 C x + 3 C$

Step 1.1.7

Simplify the expression.

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Step 1.1.7.1

Move $B$ .

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + 2 x^{2} B + 3 B x + 2 C x + 3 C$

Step 1.1.7.2

Move $B$ .

$2 x^{2} + 2 x + 11 = A x^{2} + 4 A + 2 x^{2} B + 3 x B + 2 C x + 3 C$

Step 1.1.7.3

Move $4 A$ .

$2 x^{2} + 2 x + 11 = A x^{2} + 2 x^{2} B + 3 x B + 2 C x + 4 A + 3 C$

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.

$2 = A + 2 B$

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.

$2 = 3 B + 2 C$

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.

$11 = 4 A + 3 C$

Step 1.2.4

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

$2 = A + 2 B$

$2 = 3 B + 2 C$

$11 = 4 A + 3 C$

$2 = A + 2 B$

$2 = 3 B + 2 C$

$11 = 4 A + 3 C$

Step 1.3

Solve the system of equations.

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Step 1.3.1

Solve for $A$ in $2 = A + 2 B$ .

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Step 1.3.1.1

Rewrite the equation as $A + 2 B = 2$ .

$A + 2 B = 2$

$2 = 3 B + 2 C$

$11 = 4 A + 3 C$

Step 1.3.1.2

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

$A = 2 - 2 B$

$2 = 3 B + 2 C$

$11 = 4 A + 3 C$

$A = 2 - 2 B$

$2 = 3 B + 2 C$

$11 = 4 A + 3 C$

Step 1.3.2

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

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Step 1.3.2.1

Replace all occurrences of $A$ in $11 = 4 A + 3 C$ with $2 - 2 B$ .

$11 = 4 (2 - 2 B) + 3 C$

$A = 2 - 2 B$

$2 = 3 B + 2 C$

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.

$11 = 4 \cdot 2 + 4 (- 2 B) + 3 C$

$A = 2 - 2 B$

$2 = 3 B + 2 C$

Step 1.3.2.2.1.2

Multiply $4$ by $2$ .

$11 = 8 + 4 (- 2 B) + 3 C$

$A = 2 - 2 B$

$2 = 3 B + 2 C$

Step 1.3.2.2.1.3

Multiply $- 2$ by $4$ .

$11 = 8 - 8 B + 3 C$

$A = 2 - 2 B$

$2 = 3 B + 2 C$

$11 = 8 - 8 B + 3 C$

$A = 2 - 2 B$

$2 = 3 B + 2 C$

$11 = 8 - 8 B + 3 C$

$A = 2 - 2 B$

$2 = 3 B + 2 C$

$11 = 8 - 8 B + 3 C$

$A = 2 - 2 B$

$2 = 3 B + 2 C$

Step 1.3.3

Reorder $2$ and $- 2 B$ .

$A = - 2 B + 2$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

Step 1.3.4

Solve for $B$ in $A = - 2 B + 2$ .

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Step 1.3.4.1

Rewrite the equation as $- 2 B + 2 = A$ .

$- 2 B + 2 = A$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

Step 1.3.4.2

Subtract $2$ from both sides of the equation.

$- 2 B = A - 2$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

Step 1.3.4.3

Divide each term in $- 2 B = A - 2$ by $- 2$ and simplify.

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Step 1.3.4.3.1

Divide each term in $- 2 B = A - 2$ by $- 2$ .

$\frac{- 2 B}{- 2} = \frac{A}{- 2} + \frac{- 2}{- 2}$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

Step 1.3.4.3.2

Simplify the left side.

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Step 1.3.4.3.2.1

Cancel the common factor of $- 2$ .

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Step 1.3.4.3.2.1.1

Cancel the common factor.

$\frac{- 2 B}{- 2} = \frac{A}{- 2} + \frac{- 2}{- 2}$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

Step 1.3.4.3.2.1.2

Divide $B$ by $1$ .

$B = \frac{A}{- 2} + \frac{- 2}{- 2}$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

$B = \frac{A}{- 2} + \frac{- 2}{- 2}$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

$B = \frac{A}{- 2} + \frac{- 2}{- 2}$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

Step 1.3.4.3.3

Simplify the right side.

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Step 1.3.4.3.3.1

Simplify each term.

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Step 1.3.4.3.3.1.1

Move the negative in front of the fraction.

$B = - \frac{A}{2} + \frac{- 2}{- 2}$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

Step 1.3.4.3.3.1.2

Divide $- 2$ by $- 2$ .

$B = - \frac{A}{2} + 1$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

$B = - \frac{A}{2} + 1$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

$B = - \frac{A}{2} + 1$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

$B = - \frac{A}{2} + 1$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

$B = - \frac{A}{2} + 1$

$11 = 8 - 8 B + 3 C$

$2 = 3 B + 2 C$

Step 1.3.5

Replace all occurrences of $B$ with $- \frac{A}{2} + 1$ in each equation.

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Step 1.3.5.1

Replace all occurrences of $B$ in $11 = 8 - 8 B + 3 C$ with $- \frac{A}{2} + 1$ .

$11 = 8 - 8 (- \frac{A}{2} + 1) + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2

Simplify the right side.

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Step 1.3.5.2.1

Simplify $8 - 8 (- \frac{A}{2} + 1) + 3 C$ .

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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.

$11 = 8 - 8 (- \frac{A}{2}) - 8 \cdot 1 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.3.5.2.1.1.2.1

Move the leading negative in $- \frac{A}{2}$ into the numerator.

$11 = 8 - 8 \frac{- A}{2} - 8 \cdot 1 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2.1.1.2.2

Factor $2$ out of $- 8$ .

$11 = 8 + 2 (- 4) (\frac{- A}{2}) - 8 \cdot 1 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2.1.1.2.3

Cancel the common factor.

$11 = 8 + 2 \cdot (- 4 \frac{- A}{2}) - 8 \cdot 1 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2.1.1.2.4

Rewrite the expression.

$11 = 8 - 4 (- A) - 8 \cdot 1 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

$11 = 8 - 4 (- A) - 8 \cdot 1 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2.1.1.3

Multiply $- 1$ by $- 4$ .

$11 = 8 + 4 A - 8 \cdot 1 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2.1.1.4

Multiply $- 8$ by $1$ .

$11 = 8 + 4 A - 8 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

$11 = 8 + 4 A - 8 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2.1.2

Combine the opposite terms in $8 + 4 A - 8 + 3 C$ .

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Step 1.3.5.2.1.2.1

Subtract $8$ from $8$ .

$11 = 4 A + 0 + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.2.1.2.2

Add $4 A$ and $0$ .

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 = 3 B + 2 C$

Step 1.3.5.3

Replace all occurrences of $B$ in $2 = 3 B + 2 C$ with $- \frac{A}{2} + 1$ .

$2 = 3 (- \frac{A}{2} + 1) + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.5.4

Simplify the right side.

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Step 1.3.5.4.1

Simplify each term.

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Step 1.3.5.4.1.1

Apply the distributive property.

$2 = 3 (- \frac{A}{2}) + 3 \cdot 1 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.5.4.1.2

Multiply $3 (- \frac{A}{2})$ .

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Step 1.3.5.4.1.2.1

Multiply $- 1$ by $3$ .

$2 = - 3 \frac{A}{2} + 3 \cdot 1 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.5.4.1.2.2

Combine $- 3$ and $\frac{A}{2}$ .

$2 = \frac{- 3 A}{2} + 3 \cdot 1 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 = \frac{- 3 A}{2} + 3 \cdot 1 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.5.4.1.3

Multiply $3$ by $1$ .

$2 = \frac{- 3 A}{2} + 3 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.5.4.1.4

Move the negative in front of the fraction.

$2 = - \frac{3 A}{2} + 3 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 = - \frac{3 A}{2} + 3 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 = - \frac{3 A}{2} + 3 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 = - \frac{3 A}{2} + 3 + 2 C$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6

Solve for $C$ in $2 = - \frac{3 A}{2} + 3 + 2 C$ .

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Step 1.3.6.1

Rewrite the equation as $- \frac{3 A}{2} + 3 + 2 C = 2$ .

$- \frac{3 A}{2} + 3 + 2 C = 2$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.2

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

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Step 1.3.6.2.1

Add $\frac{3 A}{2}$ to both sides of the equation.

$3 + 2 C = 2 + \frac{3 A}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.2.2

Subtract $3$ from both sides of the equation.

$2 C = 2 + \frac{3 A}{2} - 3$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.2.3

Subtract $3$ from $2$ .

$2 C = \frac{3 A}{2} - 1$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$2 C = \frac{3 A}{2} - 1$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.3

Divide each term in $2 C = \frac{3 A}{2} - 1$ by $2$ and simplify.

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Step 1.3.6.3.1

Divide each term in $2 C = \frac{3 A}{2} - 1$ by $2$ .

$\frac{2 C}{2} = \frac{\frac{3 A}{2}}{2} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

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 $2$ .

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Step 1.3.6.3.2.1.1

Cancel the common factor.

$\frac{2 C}{2} = \frac{\frac{3 A}{2}}{2} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{\frac{3 A}{2}}{2} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$C = \frac{\frac{3 A}{2}}{2} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$C = \frac{\frac{3 A}{2}}{2} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.3.3

Simplify the right side.

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Step 1.3.6.3.3.1

Simplify each term.

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Step 1.3.6.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$C = \frac{3 A}{2} \cdot \frac{1}{2} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.3.3.1.2

Multiply $\frac{3 A}{2} \cdot \frac{1}{2}$ .

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Step 1.3.6.3.3.1.2.1

Multiply $\frac{3 A}{2}$ by $\frac{1}{2}$ .

$C = \frac{3 A}{2 \cdot 2} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.3.3.1.2.2

Multiply $2$ by $2$ .

$C = \frac{3 A}{4} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$C = \frac{3 A}{4} + \frac{- 1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.6.3.3.1.3

Move the negative in front of the fraction.

$C = \frac{3 A}{4} - \frac{1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$C = \frac{3 A}{4} - \frac{1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$C = \frac{3 A}{4} - \frac{1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$C = \frac{3 A}{4} - \frac{1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

$C = \frac{3 A}{4} - \frac{1}{2}$

$11 = 4 A + 3 C$

$B = - \frac{A}{2} + 1$

Step 1.3.7

Replace all occurrences of $C$ with $\frac{3 A}{4} - \frac{1}{2}$ in each equation.

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Step 1.3.7.1

Replace all occurrences of $C$ in $11 = 4 A + 3 C$ with $\frac{3 A}{4} - \frac{1}{2}$ .

$11 = 4 A + 3 (\frac{3 A}{4} - \frac{1}{2})$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2

Simplify the right side.

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Step 1.3.7.2.1

Simplify $4 A + 3 (\frac{3 A}{4} - \frac{1}{2})$ .

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Step 1.3.7.2.1.1

Simplify each term.

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Step 1.3.7.2.1.1.1

Apply the distributive property.

$11 = 4 A + 3 (\frac{3 A}{4}) + 3 (- \frac{1}{2})$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.1.2

Multiply $3 \frac{3 A}{4}$ .

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Step 1.3.7.2.1.1.2.1

Combine $3$ and $\frac{3 A}{4}$ .

$11 = 4 A + \frac{3 (3 A)}{4} + 3 (- \frac{1}{2})$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.1.2.2

Multiply $3$ by $3$ .

$11 = 4 A + \frac{9 A}{4} + 3 (- \frac{1}{2})$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = 4 A + \frac{9 A}{4} + 3 (- \frac{1}{2})$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.1.3

Multiply $3 (- \frac{1}{2})$ .

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Step 1.3.7.2.1.1.3.1

Multiply $- 1$ by $3$ .

$11 = 4 A + \frac{9 A}{4} - 3 (\frac{1}{2})$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.1.3.2

Combine $- 3$ and $\frac{1}{2}$ .

$11 = 4 A + \frac{9 A}{4} + \frac{- 3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = 4 A + \frac{9 A}{4} + \frac{- 3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.1.4

Move the negative in front of the fraction.

$11 = 4 A + \frac{9 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = 4 A + \frac{9 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.2

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

$11 = 4 A \cdot \frac{4}{4} + \frac{9 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.3

Simplify terms.

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Step 1.3.7.2.1.3.1

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

$11 = \frac{4 A \cdot 4}{4} + \frac{9 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.3.2

Combine the numerators over the common denominator.

$11 = \frac{4 A \cdot 4 + 9 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = \frac{4 A \cdot 4 + 9 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.4

Simplify each term.

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Step 1.3.7.2.1.4.1

Simplify the numerator.

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Step 1.3.7.2.1.4.1.1

Factor $A$ out of $4 A \cdot 4 + 9 A$ .

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Step 1.3.7.2.1.4.1.1.1

Factor $A$ out of $4 A \cdot 4$ .

$11 = \frac{A (4 \cdot 4) + 9 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.4.1.1.2

Factor $A$ out of $9 A$ .

$11 = \frac{A (4 \cdot 4) + A \cdot 9}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.4.1.1.3

Factor $A$ out of $A (4 \cdot 4) + A \cdot 9$ .

$11 = \frac{A (4 \cdot 4 + 9)}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = \frac{A (4 \cdot 4 + 9)}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.4.1.2

Multiply $4$ by $4$ .

$11 = \frac{A (16 + 9)}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.4.1.3

Add $16$ and $9$ .

$11 = \frac{A \cdot 25}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = \frac{A \cdot 25}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.7.2.1.4.2

Move $25$ to the left of $A$ .

$11 = \frac{25 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = \frac{25 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = \frac{25 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = \frac{25 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$11 = \frac{25 A}{4} - \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8

Solve for $A$ in $11 = \frac{25 A}{4} - \frac{3}{2}$ .

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Step 1.3.8.1

Rewrite the equation as $\frac{25 A}{4} - \frac{3}{2} = 11$ .

$\frac{25 A}{4} - \frac{3}{2} = 11$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.2

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

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Step 1.3.8.2.1

Add $\frac{3}{2}$ to both sides of the equation.

$\frac{25 A}{4} = 11 + \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.2.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{25 A}{4} = 11 \cdot \frac{2}{2} + \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.2.3

Combine $11$ and $\frac{2}{2}$ .

$\frac{25 A}{4} = \frac{11 \cdot 2}{2} + \frac{3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.2.4

Combine the numerators over the common denominator.

$\frac{25 A}{4} = \frac{11 \cdot 2 + 3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.2.5

Simplify the numerator.

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Step 1.3.8.2.5.1

Multiply $11$ by $2$ .

$\frac{25 A}{4} = \frac{22 + 3}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.2.5.2

Add $22$ and $3$ .

$\frac{25 A}{4} = \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$\frac{25 A}{4} = \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$\frac{25 A}{4} = \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.3

Multiply both sides of the equation by $\frac{4}{25}$ .

$\frac{4}{25} \cdot \frac{25 A}{4} = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4

Simplify both sides of the equation.

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Step 1.3.8.4.1

Simplify the left side.

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Step 1.3.8.4.1.1

Simplify $\frac{4}{25} \cdot \frac{25 A}{4}$ .

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Step 1.3.8.4.1.1.1

Cancel the common factor of $4$ .

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Step 1.3.8.4.1.1.1.1

Cancel the common factor.

$\frac{4}{25} \cdot \frac{25 A}{4} = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.1.1.1.2

Rewrite the expression.

$\frac{1}{25} \cdot (25 A) = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$\frac{1}{25} \cdot (25 A) = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.1.1.2

Cancel the common factor of $25$ .

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Step 1.3.8.4.1.1.2.1

Factor $25$ out of $25 A$ .

$\frac{1}{25} \cdot (25 (A)) = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.1.1.2.2

Cancel the common factor.

$\frac{1}{25} \cdot (25 A) = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.1.1.2.3

Rewrite the expression.

$A = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = \frac{4}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.2

Simplify the right side.

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Step 1.3.8.4.2.1

Simplify $\frac{4}{25} \cdot \frac{25}{2}$ .

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Step 1.3.8.4.2.1.1

Cancel the common factor of $2$ .

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Step 1.3.8.4.2.1.1.1

Factor $2$ out of $4$ .

$A = \frac{2 (2)}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.2.1.1.2

Cancel the common factor.

$A = \frac{2 \cdot 2}{25} \cdot \frac{25}{2}$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.2.1.1.3

Rewrite the expression.

$A = \frac{2}{25} \cdot 25$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = \frac{2}{25} \cdot 25$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.2.1.2

Cancel the common factor of $25$ .

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Step 1.3.8.4.2.1.2.1

Cancel the common factor.

$A = \frac{2}{25} \cdot 25$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.8.4.2.1.2.2

Rewrite the expression.

$A = 2$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = 2$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = 2$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = 2$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = 2$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

$A = 2$

$C = \frac{3 A}{4} - \frac{1}{2}$

$B = - \frac{A}{2} + 1$

Step 1.3.9

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

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Step 1.3.9.1

Replace all occurrences of $A$ in $C = \frac{3 A}{4} - \frac{1}{2}$ with $2$ .

$C = \frac{3 (2)}{4} - \frac{1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.2

Simplify the right side.

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Step 1.3.9.2.1

Simplify $\frac{3 (2)}{4} - \frac{1}{2}$ .

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Step 1.3.9.2.1.1

Multiply $3$ by $2$ .

$C = \frac{6}{4} - \frac{1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.2.1.2

Cancel the common factor of $6$ and $4$ .

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Step 1.3.9.2.1.2.1

Factor $2$ out of $6$ .

$C = \frac{2 (3)}{4} - \frac{1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.2.1.2.2

Cancel the common factors.

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Step 1.3.9.2.1.2.2.1

Factor $2$ out of $4$ .

$C = \frac{2 \cdot 3}{2 \cdot 2} - \frac{1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.2.1.2.2.2

Cancel the common factor.

$C = \frac{2 \cdot 3}{2 \cdot 2} - \frac{1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.2.1.2.2.3

Rewrite the expression.

$C = \frac{3}{2} - \frac{1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

$C = \frac{3}{2} - \frac{1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

$C = \frac{3}{2} - \frac{1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.2.1.3

Combine the numerators over the common denominator.

$C = \frac{3 - 1}{2}$

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.2.1.4

Simplify the expression.

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Step 1.3.9.2.1.4.1

Subtract $1$ from $3$ .

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

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.2.1.4.2

Divide $2$ by $2$ .

$C = 1$

$A = 2$

$B = - \frac{A}{2} + 1$

$C = 1$

$A = 2$

$B = - \frac{A}{2} + 1$

$C = 1$

$A = 2$

$B = - \frac{A}{2} + 1$

$C = 1$

$A = 2$

$B = - \frac{A}{2} + 1$

Step 1.3.9.3

Replace all occurrences of $A$ in $B = - \frac{A}{2} + 1$ with $2$ .

$B = - \frac{2}{2} + 1$

$C = 1$

$A = 2$

Step 1.3.9.4

Simplify the right side.

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Step 1.3.9.4.1

Simplify $- \frac{2}{2} + 1$ .

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Step 1.3.9.4.1.1

Simplify each term.

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Step 1.3.9.4.1.1.1

Cancel the common factor of $2$ .

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Step 1.3.9.4.1.1.1.1

Cancel the common factor.

$B = - \frac{2}{2} + 1$

$C = 1$

$A = 2$

Step 1.3.9.4.1.1.1.2

Rewrite the expression.

$B = - 1 \cdot 1 + 1$

$C = 1$

$A = 2$

$B = - 1 \cdot 1 + 1$

$C = 1$

$A = 2$

Step 1.3.9.4.1.1.2

Multiply $- 1$ by $1$ .

$B = - 1 + 1$

$C = 1$

$A = 2$

$B = - 1 + 1$

$C = 1$

$A = 2$

Step 1.3.9.4.1.2

Add $- 1$ and $1$ .

$B = 0$

$C = 1$

$A = 2$

$B = 0$

$C = 1$

$A = 2$

$B = 0$

$C = 1$

$A = 2$

$B = 0$

$C = 1$

$A = 2$

Step 1.3.10

List all of the solutions.

$B = 0, C = 1, A = 2$

Step 1.4

Replace each of the partial fraction coefficients in $\frac{A}{2 x + 3} + \frac{B x + C}{x^{2} + 4}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{2}{2 x + 3} + \frac{0 x + 1}{x^{2} + 4}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $0$ by $x$ .

$\frac{2}{2 x + 3} + \frac{0 + 1}{x^{2} + 4}$

Step 1.5.2

Add $0$ and $1$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Let $u = 2 x + 3$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 2 x + 3$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $2 x + 3$ .

$\frac{d}{d x} [2 x + 3]$

Step 4.1.2

By the Sum Rule, the derivative of $2 x + 3$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [2 x] + \frac{d}{d x} [3]$

Step 4.1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 4.1.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 4.1.3.2

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

$2 \cdot 1 + \frac{d}{d x} [3]$

Step 4.1.3.3

Multiply $2$ by $1$ .

$2 + \frac{d}{d x} [3]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

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

$2 + 0$

Step 4.1.4.2

Add $2$ and $0$ .

$2$

Step 4.2

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

$2 \int \frac{1}{u} \cdot \frac{1}{2} d u + \int \frac{1}{x^{2} + 4} d x$

Step 5

Simplify.

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Step 5.1

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

$2 \int \frac{1}{u \cdot 2} d u + \int \frac{1}{x^{2} + 4} d x$

Step 5.2

Move $2$ to the left of $u$ .

$2 \int \frac{1}{2 u} d u + \int \frac{1}{x^{2} + 4} d x$

Step 6

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

$2 (\frac{1}{2} \int \frac{1}{u} d u) + \int \frac{1}{x^{2} + 4} d x$

Step 7

Simplify.

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Step 7.1

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

$\frac{2}{2} \int \frac{1}{u} d u + \int \frac{1}{x^{2} + 4} d x$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Cancel the common factor.

$\frac{2}{2} \int \frac{1}{u} d u + \int \frac{1}{x^{2} + 4} d x$

Step 7.2.2

Rewrite the expression.

$1 \int \frac{1}{u} d u + \int \frac{1}{x^{2} + 4} d x$

Step 7.3

Multiply $\int \frac{1}{u} d u$ by $1$ .

$\int \frac{1}{u} d u + \int \frac{1}{x^{2} + 4} d x$

Step 8

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

$\ln (| u |) + C + \int \frac{1}{x^{2} + 4} d x$

Step 9

Simplify the expression.

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Step 9.1

Reorder $x^{2}$ and $4$ .

$\ln (| u |) + C + \int \frac{1}{4 + x^{2}} d x$

Step 9.2

Rewrite $4$ as $2^{2}$ .

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{2}$ and $\arctan (\frac{x}{2})$ .

$\ln (| u |) + C + \frac{\arctan (\frac{x}{2})}{2} + C$

Step 11.2

Simplify.

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

Step 12

Replace all occurrences of $u$ with $2 x + 3$ .

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