Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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

Step 3

Let $u_{1} = x - 4$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = x - 4$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $x - 4$ .

$\frac{d}{d x} [x - 4]$

Step 3.1.2

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

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

Step 3.1.3

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

$1 + \frac{d}{d x} [- 4]$

Step 3.1.4

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

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

Step 3.2

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Multiply $3$ by $- 1$ .

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

Step 8

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

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Step 8.1

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

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Step 8.1.1

Differentiate $2 x + 1$ .

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

Step 8.1.2

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

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

Step 8.1.3

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

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Step 8.1.3.1

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

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

Step 8.1.3.2

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

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

Step 8.1.3.3

Multiply $2$ by $1$ .

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

Step 8.1.4

Differentiate using the Constant Rule.

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Step 8.1.4.1

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

$2 + 0$

Step 8.1.4.2

Add $2$ and $0$ .

$2$

Step 8.2

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

Simplify.

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

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of $u_{1}$ with $x - 4$ .

$2 \ln (| x - 4 |) - \frac{3}{2} \ln (| u_{2} |) + C$

Step 14.2

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

$2 \ln (| x - 4 |) - \frac{3}{2} \ln (| 2 x + 1 |) + C$