Calculus Examples

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

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.

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

Step 1.1.2

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

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

Step 1.1.3

Cancel the common factor of $x$ .

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Step 1.1.3.1

Cancel the common factor.

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

Step 1.1.3.2

Rewrite the expression.

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

Step 1.1.4

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

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Step 1.1.4.1

Cancel the common factor.

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

Step 1.1.4.2

Rewrite the expression.

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

Step 1.1.5

Simplify each term.

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Step 1.1.5.1

Cancel the common factor of $x$ .

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Step 1.1.5.1.1

Cancel the common factor.

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

Step 1.1.5.1.2

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

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

Step 1.1.5.2

Apply the distributive property.

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

Step 1.1.5.3

Multiply $A$ by $1$ .

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

Step 1.1.5.4

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

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Step 1.1.5.4.1

Cancel the common factor.

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

Step 1.1.5.4.2

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

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

Step 1.1.5.5

Apply the distributive property.

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

Step 1.1.5.6

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

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Step 1.1.5.6.1

Move $x$ .

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

Step 1.1.5.6.2

Multiply $x$ by $x$ .

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

Step 1.1.6

Move $A$ .

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

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.

$0 = A + B$

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.

$0 = C$

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.

$1 = A$

Step 1.2.4

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 1.3

Solve the system of equations.

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Step 1.3.1

Rewrite the equation as $C = 0$ .

$C = 0$

$0 = A + B$

$1 = A$

Step 1.3.2

Rewrite the equation as $A = 1$ .

$A = 1$

$C = 0$

$0 = A + B$

Step 1.3.3

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

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Step 1.3.3.1

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

$0 = (1) + B$

$A = 1$

$C = 0$

Step 1.3.3.2

Simplify the right side.

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Step 1.3.3.2.1

Remove parentheses.

$0 = 1 + B$

$A = 1$

$C = 0$

$0 = 1 + B$

$A = 1$

$C = 0$

$0 = 1 + B$

$A = 1$

$C = 0$

Step 1.3.4

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

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Step 1.3.4.1

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

$1 + B = 0$

$A = 1$

$C = 0$

Step 1.3.4.2

Subtract $1$ from both sides of the equation.

$B = - 1$

$A = 1$

$C = 0$

$B = - 1$

$A = 1$

$C = 0$

Step 1.3.5

Solve the system of equations.

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

Step 1.3.6

List all of the solutions.

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

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Remove parentheses.

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

Step 1.5.2

Simplify the numerator.

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Step 1.5.2.1

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

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

Step 1.5.2.2

Add $- x$ and $0$ .

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

Step 1.5.3

Move the negative in front of the fraction.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Let $u = 1 + x^{2}$ . 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 5.1

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

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Step 5.1.1

Differentiate $1 + x^{2}$ .

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

Step 5.1.2

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

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

Step 5.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{2}]$

Step 5.1.4

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

$0 + 2 x$

Step 5.1.5

Add $0$ and $2 x$ .

$2 x$

Step 5.2

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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 8

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

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

Step 9

Simplify.

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

Step 10

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

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