Calculus Examples

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

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 2

Write the fraction using partial fraction decomposition.

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Step 2.1

Decompose the fraction and multiply through by the common denominator.

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Step 2.1.1

Factor $x$ out of $x^{3} + 4 x$ .

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Step 2.1.1.1

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

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

Step 2.1.1.2

Factor $x$ out of $4 x$ .

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

Step 2.1.1.3

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

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

Step 2.1.2

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.

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

Step 2.1.3

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

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

Step 2.1.4

Cancel the common factor of $x$ .

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Step 2.1.4.1

Cancel the common factor.

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

Step 2.1.4.2

Rewrite the expression.

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

Step 2.1.5

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

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Step 2.1.5.1

Cancel the common factor.

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

Step 2.1.5.2

Rewrite the expression.

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

Step 2.1.6

Simplify each term.

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Step 2.1.6.1

Cancel the common factor of $x$ .

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Step 2.1.6.1.1

Cancel the common factor.

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

Step 2.1.6.1.2

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

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

Step 2.1.6.2

Apply the distributive property.

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

Step 2.1.6.3

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

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

Step 2.1.6.4

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

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Step 2.1.6.4.1

Cancel the common factor.

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

Step 2.1.6.4.2

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

$1 = A x^{2} + 4 A + (B x + C) (x)$

Step 2.1.6.5

Apply the distributive property.

$1 = A x^{2} + 4 A + B x \cdot x + C x$

Step 2.1.6.6

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

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Step 2.1.6.6.1

Move $x$ .

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

Step 2.1.6.6.2

Multiply $x$ by $x$ .

$1 = A x^{2} + 4 A + B x^{2} + C x$

Step 2.1.7

Move $4 A$ .

$1 = A x^{2} + B x^{2} + C x + 4 A$

Step 2.2

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

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Step 2.2.1

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.

$0 = A + B$

Step 2.2.2

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.

$0 = C$

Step 2.2.3

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.

$1 = 4 A$

Step 2.2.4

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

$0 = A + B$

$0 = C$

$1 = 4 A$

$0 = A + B$

$0 = C$

$1 = 4 A$

Step 2.3

Solve the system of equations.

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Step 2.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$0 = A + B$

$1 = 4 A$

Step 2.3.2

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

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Step 2.3.2.1

Rewrite the equation as $4 A = 1$ .

$4 A = 1$

$C = 0$

$0 = A + B$

Step 2.3.2.2

Divide each term in $4 A = 1$ by $4$ and simplify.

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Step 2.3.2.2.1

Divide each term in $4 A = 1$ by $4$ .

$\frac{4 A}{4} = \frac{1}{4}$

$C = 0$

$0 = A + B$

Step 2.3.2.2.2

Simplify the left side.

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Step 2.3.2.2.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.2.2.1.1

Cancel the common factor.

$\frac{4 A}{4} = \frac{1}{4}$

$C = 0$

$0 = A + B$

Step 2.3.2.2.2.1.2

Divide $A$ by $1$ .

$A = \frac{1}{4}$

$C = 0$

$0 = A + B$

$A = \frac{1}{4}$

$C = 0$

$0 = A + B$

$A = \frac{1}{4}$

$C = 0$

$0 = A + B$

$A = \frac{1}{4}$

$C = 0$

$0 = A + B$

$A = \frac{1}{4}$

$C = 0$

$0 = A + B$

Step 2.3.3

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

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Step 2.3.3.1

Replace all occurrences of $A$ in $0 = A + B$ with $\frac{1}{4}$ .

$0 = (\frac{1}{4}) + B$

$A = \frac{1}{4}$

$C = 0$

Step 2.3.3.2

Simplify the right side.

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Step 2.3.3.2.1

Remove parentheses.

$0 = \frac{1}{4} + B$

$A = \frac{1}{4}$

$C = 0$

$0 = \frac{1}{4} + B$

$A = \frac{1}{4}$

$C = 0$

$0 = \frac{1}{4} + B$

$A = \frac{1}{4}$

$C = 0$

Step 2.3.4

Solve for $B$ in $0 = \frac{1}{4} + B$ .

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Step 2.3.4.1

Rewrite the equation as $\frac{1}{4} + B = 0$ .

$\frac{1}{4} + B = 0$

$A = \frac{1}{4}$

$C = 0$

Step 2.3.4.2

Subtract $\frac{1}{4}$ from both sides of the equation.

$B = - \frac{1}{4}$

$A = \frac{1}{4}$

$C = 0$

$B = - \frac{1}{4}$

$A = \frac{1}{4}$

$C = 0$

Step 2.3.5

Solve the system of equations.

$B = - \frac{1}{4} A = \frac{1}{4} C = 0$

Step 2.3.6

List all of the solutions.

$B = - \frac{1}{4}, A = \frac{1}{4}, C = 0$

Step 2.4

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

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

Step 2.5

Simplify.

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Step 2.5.1

Remove parentheses.

$\frac{\frac{1}{4}}{x} + \frac{(- \frac{1}{4}) \cdot x + 0}{x^{2} + 4}$

Step 2.5.2

Simplify the numerator.

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Step 2.5.2.1

Combine $x$ and $\frac{1}{4}$ .

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

Step 2.5.2.2

Add $- \frac{x}{4}$ and $0$ .

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

Step 2.5.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.4

Multiply $\frac{1}{x^{2} + 4}$ by $\frac{x}{4}$ .

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

Step 2.5.5

Move $4$ to the left of $x^{2} + 4$ .

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

Step 2.5.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.7

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

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

Step 5

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

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

Step 6

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

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

Step 7

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

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

Step 8

Let $u = x^{2} + 4$ . 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 8.1

Let $u = x^{2} + 4$ . Find $\frac{d u}{d x}$ .

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Step 8.1.1

Differentiate $x^{2} + 4$ .

$\frac{d}{d x} [x^{2} + 4]$

Step 8.1.2

By the Sum Rule, the derivative of $x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]$

Step 8.1.3

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

$2 x + \frac{d}{d x} [4]$

Step 8.1.4

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

$2 x + 0$

Step 8.1.5

Add $2 x$ and $0$ .

$2 x$

Step 8.2

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

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

Step 11.2

Multiply $2$ by $4$ .

$4 (\frac{1}{4} (\ln (| x |) + C) - \frac{1}{8} \int \frac{1}{u} d u)$

Step 12

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

$4 (\frac{1}{4} (\ln (| x |) + C) - \frac{1}{8} (\ln (| u |) + C))$

Step 13

Simplify.

$4 (\frac{1}{4} \ln (| x |) - \frac{1}{8} \ln (| u |)) + C$

Step 14

Replace all occurrences of $u$ with $x^{2} + 4$ .

$4 (\frac{1}{4} \ln (| x |) - \frac{1}{8} \ln (| x^{2} + 4 |)) + C$

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

Combine $\frac{1}{4}$ and $\ln (| x |)$ .

$4 (\frac{\ln (| x |)}{4} - \frac{1}{8} \ln (| x^{2} + 4 |)) + C$

Step 15.1.2

Combine $\ln (| x^{2} + 4 |)$ and $\frac{1}{8}$ .

$4 (\frac{\ln (| x |)}{4} - \frac{\ln (| x^{2} + 4 |)}{8}) + C$

Step 15.2

To write $\frac{\ln (| x |)}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 (\frac{\ln (| x |)}{4} \cdot \frac{2}{2} - \frac{\ln (| x^{2} + 4 |)}{8}) + C$

Step 15.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 15.3.1

Multiply $\frac{\ln (| x |)}{4}$ by $\frac{2}{2}$ .

$4 (\frac{\ln (| x |) \cdot 2}{4 \cdot 2} - \frac{\ln (| x^{2} + 4 |)}{8}) + C$

Step 15.3.2

Multiply $4$ by $2$ .

$4 (\frac{\ln (| x |) \cdot 2}{8} - \frac{\ln (| x^{2} + 4 |)}{8}) + C$

Step 15.4

Combine the numerators over the common denominator.

$4 \frac{\ln (| x |) \cdot 2 - \ln (| x^{2} + 4 |)}{8} + C$

Step 15.5

Cancel the common factor of $4$ .

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Step 15.5.1

Factor $4$ out of $8$ .

$4 \frac{\ln (| x |) \cdot 2 - \ln (| x^{2} + 4 |)}{4 (2)} + C$

Step 15.5.2

Cancel the common factor.

$4 \frac{\ln (| x |) \cdot 2 - \ln (| x^{2} + 4 |)}{4 \cdot 2} + C$

Step 15.5.3

Rewrite the expression.

$\frac{\ln (| x |) \cdot 2 - \ln (| x^{2} + 4 |)}{2} + C$

Step 15.6

Move $2$ to the left of $\ln (| x |)$ .

$\frac{2 \ln (| x |) - \ln (| x^{2} + 4 |)}{2} + C$

Step 16

Reorder terms.

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