Calculus Examples

$\frac{1}{X^{3} + X}$

Step 1

Write $\frac{1}{X^{3} + X}$ as a function.

$f (X) = \frac{1}{X^{3} + X}$

Step 2

The function $F (X)$ can be found by finding the indefinite integral of the derivative $f (X)$ .

$F (X) = \int f (X) d X$

Step 3

Set up the integral to solve.

$F (X) = \int \frac{1}{X^{3} + X} d X$

Step 4

Write the fraction using partial fraction decomposition.

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Step 4.1

Decompose the fraction and multiply through by the common denominator.

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Step 4.1.1

Factor $X$ out of $X^{3} + X$ .

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Step 4.1.1.1

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

$\frac{1}{X \cdot X^{2} + X}$

Step 4.1.1.2

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

$\frac{1}{X \cdot X^{2} + X^{1}}$

Step 4.1.1.3

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

$\frac{1}{X \cdot X^{2} + X \cdot 1}$

Step 4.1.1.4

Factor $X$ out of $X \cdot X^{2} + X \cdot 1$ .

$\frac{1}{X (X^{2} + 1)}$

Step 4.1.2

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.

$\frac{A}{X} + \frac{B X + C}{X^{2} + 1}$

Step 4.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $X (X^{2} + 1)$ .

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

Step 4.1.4

Cancel the common factor of $X$ .

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Step 4.1.4.1

Cancel the common factor.

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

Step 4.1.4.2

Rewrite the expression.

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

Step 4.1.5

Cancel the common factor of $X^{2} + 1$ .

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Step 4.1.5.1

Cancel the common factor.

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

Step 4.1.5.2

Rewrite the expression.

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

Step 4.1.6

Simplify each term.

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Step 4.1.6.1

Cancel the common factor of $X$ .

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Step 4.1.6.1.1

Cancel the common factor.

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

Step 4.1.6.1.2

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

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

Step 4.1.6.2

Apply the distributive property.

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

Step 4.1.6.3

Multiply $A$ by $1$ .

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

Step 4.1.6.4

Cancel the common factor of $X^{2} + 1$ .

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Step 4.1.6.4.1

Cancel the common factor.

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

Step 4.1.6.4.2

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

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

Step 4.1.6.5

Apply the distributive property.

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

Step 4.1.6.6

Multiply $X$ by $X$ by adding the exponents.

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Step 4.1.6.6.1

Move $X$ .

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

Step 4.1.6.6.2

Multiply $X$ by $X$ .

$1 = A X^{2} + A + B X^{2} + C X$

Step 4.1.7

Move $A$ .

$1 = A X^{2} + B X^{2} + C X + A$

Step 4.2

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

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Step 4.2.1

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 4.2.2

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 4.2.3

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.

$1 = A$

Step 4.2.4

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

$0 = A + B$

$0 = C$

$1 = A$

$0 = A + B$

$0 = C$

$1 = A$

Step 4.3

Solve the system of equations.

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Step 4.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$0 = A + B$

$1 = A$

Step 4.3.2

Rewrite the equation as $A = 1$ .

$A = 1$

$C = 0$

$0 = A + B$

Step 4.3.3

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

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Step 4.3.3.1

Replace all occurrences of $A$ in $0 = A + B$ with $1$ .

$0 = (1) + B$

$A = 1$

$C = 0$

Step 4.3.3.2

Simplify the right side.

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Step 4.3.3.2.1

Remove parentheses.

$0 = 1 + B$

$A = 1$

$C = 0$

$0 = 1 + B$

$A = 1$

$C = 0$

$0 = 1 + B$

$A = 1$

$C = 0$

Step 4.3.4

Solve for $B$ in $0 = 1 + B$ .

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Step 4.3.4.1

Rewrite the equation as $1 + B = 0$ .

$1 + B = 0$

$A = 1$

$C = 0$

Step 4.3.4.2

Subtract $1$ from both sides of the equation.

$B = - 1$

$A = 1$

$C = 0$

$B = - 1$

$A = 1$

$C = 0$

Step 4.3.5

Solve the system of equations.

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

Step 4.3.6

List all of the solutions.

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

Step 4.4

Replace each of the partial fraction coefficients in $\frac{A}{X} + \frac{B X + C}{X^{2} + 1}$ with the values found for $A$ , $B$ , and $C$ .

$\frac{1}{X} + \frac{- 1 X + 0}{X^{2} + 1}$

Step 4.5

Simplify.

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Step 4.5.1

Remove parentheses.

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

Step 4.5.2

Simplify the numerator.

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Step 4.5.2.1

Rewrite $- 1 X$ as $- X$ .

$\frac{1}{X} + \frac{- X + 0}{X^{2} + 1}$

Step 4.5.2.2

Add $- X$ and $0$ .

$\frac{1}{X} + \frac{- X}{X^{2} + 1}$

Step 4.5.3

Move the negative in front of the fraction.

$\int \frac{1}{X} - \frac{X}{X^{2} + 1} d X$

Step 5

Split the single integral into multiple integrals.

$\int \frac{1}{X} d X + \int - \frac{X}{X^{2} + 1} d X$

Step 6

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

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

Step 7

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

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

Step 8

Let $u = X^{2} + 1$ . 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 $u = X^{2} + 1$ . Find $\frac{d u}{d X}$ .

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Step 8.1.1

Differentiate $X^{2} + 1$ .

$\frac{d}{d X} [X^{2} + 1]$

Step 8.1.2

By the Sum Rule, the derivative of $X^{2} + 1$ with respect to $X$ is $\frac{d}{d X} [X^{2}] + \frac{d}{d X} [1]$ .

$\frac{d}{d X} [X^{2}] + \frac{d}{d X} [1]$

Step 8.1.3

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

$2 X + \frac{d}{d X} [1]$

Step 8.1.4

Since $1$ is constant with respect to $X$ , the derivative of $1$ 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 $d u$ .

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

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

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

Step 11

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

$\ln (| X |) + C - \frac{1}{2} (\ln (| u |) + C)$

Step 12

Simplify.

$\ln (| X |) - \frac{1}{2} \ln (| u |) + C$

Step 13

Replace all occurrences of $u$ with $X^{2} + 1$ .

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

Step 14

The answer is the antiderivative of the function $f (X) = \frac{1}{X^{3} + X}$ .

$F (X) =$ $\ln (| X |) - \frac{1}{2} \ln (| X^{2} + 1 |) + C$