Calculus Examples

$\int 2 \cos (7 x) - \sin (\frac{x}{2}) - π d x$

Step 1

Split the single integral into multiple integrals.

$\int 2 \cos (7 x) d x + \int - \sin (\frac{x}{2}) d x + \int - π d x$

Step 2

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

$2 \int \cos (7 x) d x + \int - \sin (\frac{x}{2}) d x + \int - π d x$

Step 3

Let $u_{1} = 7 x$ . Then $d u_{1} = 7 d x$ , so $\frac{1}{7} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = 7 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $7 x$ .

$\frac{d}{d x} [7 x]$

Step 3.1.2

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$7 \frac{d}{d x} [x]$

Step 3.1.3

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

$7 \cdot 1$

Step 3.1.4

Multiply $7$ by $1$ .

$7$

Step 3.2

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

$2 \int \cos (u_{1}) \frac{1}{7} d u_{1} + \int - \sin (\frac{x}{2}) d x + \int - π d x$

Step 4

Combine $\cos (u_{1})$ and $\frac{1}{7}$ .

$2 \int \frac{\cos (u_{1})}{7} d u_{1} + \int - \sin (\frac{x}{2}) d x + \int - π d x$

Step 5

Since $\frac{1}{7}$ is constant with respect to $u_{1}$ , move $\frac{1}{7}$ out of the integral.

$2 (\frac{1}{7} \int \cos (u_{1}) d u_{1}) + \int - \sin (\frac{x}{2}) d x + \int - π d x$

Step 6

Combine $\frac{1}{7}$ and $2$ .

$\frac{2}{7} \int \cos (u_{1}) d u_{1} + \int - \sin (\frac{x}{2}) d x + \int - π d x$

Step 7

The integral of $\cos (u_{1})$ with respect to $u_{1}$ is $\sin (u_{1})$ .

$\frac{2}{7} (\sin (u_{1}) + C) + \int - \sin (\frac{x}{2}) d x + \int - π d x$

Step 8

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

$\frac{2}{7} (\sin (u_{1}) + C) - \int \sin (\frac{x}{2}) d x + \int - π d x$

Step 9

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

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Step 9.1

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

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Step 9.1.1

Differentiate $\frac{x}{2}$ .

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

Step 9.1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

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

Step 9.1.3

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

$\frac{1}{2} \cdot 1$

Step 9.1.4

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

$\frac{1}{2}$

Step 9.2

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

$\frac{2}{7} (\sin (u_{1}) + C) - \int \sin (u_{2}) \frac{1}{\frac{1}{2}} d u_{2} + \int - π d x$

Step 10

Simplify.

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Step 10.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\frac{2}{7} (\sin (u_{1}) + C) - \int \sin (u_{2}) (1 \cdot 2) d u_{2} + \int - π d x$

Step 10.2

Multiply $2$ by $1$ .

$\frac{2}{7} (\sin (u_{1}) + C) - \int \sin (u_{2}) \cdot 2 d u_{2} + \int - π d x$

Step 10.3

Move $2$ to the left of $\sin (u_{2})$ .

$\frac{2}{7} (\sin (u_{1}) + C) - \int 2 \sin (u_{2}) d u_{2} + \int - π d x$

Step 11

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

$\frac{2}{7} (\sin (u_{1}) + C) - (2 \int \sin (u_{2}) d u_{2}) + \int - π d x$

Step 12

Multiply $2$ by $- 1$ .

$\frac{2}{7} (\sin (u_{1}) + C) - 2 \int \sin (u_{2}) d u_{2} + \int - π d x$

Step 13

The integral of $\sin (u_{2})$ with respect to $u_{2}$ is $- \cos (u_{2})$ .

$\frac{2}{7} (\sin (u_{1}) + C) - 2 (- \cos (u_{2}) + C) + \int - π d x$

Step 14

Apply the constant rule.

$\frac{2}{7} (\sin (u_{1}) + C) - 2 (- \cos (u_{2}) + C) - π x + C$

Step 15

Simplify.

$\frac{2 \sin (u_{1})}{7} + 2 \cos (u_{2}) - π x + C$

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

Replace all occurrences of $u_{1}$ with $7 x$ .

$\frac{2 \sin (7 x)}{7} + 2 \cos (u_{2}) - π x + C$

Step 16.2

Replace all occurrences of $u_{2}$ with $\frac{x}{2}$ .

$\frac{2 \sin (7 x)}{7} + 2 \cos (\frac{x}{2}) - π x + C$

Step 17

Reorder terms.

$\frac{2}{7} \sin (7 x) + 2 \cos (\frac{1}{2} x) - π x + C$