Calculus Examples

$\int \sec^{2} (x) + 2 \sin (\frac{1}{4} x) - \cot (a n \cdot 5 x) d x$

Step 1

Simplify.

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Step 1.1

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

$\int \sec^{2} (x) + 2 \sin (\frac{x}{4}) - \cot (a n \cdot (5 x)) d x$

Step 1.2

Move $5$ to the left of $a n$ .

$\int \sec^{2} (x) + 2 \sin (\frac{x}{4}) - \cot (5 a n x) d x$

Step 2

Split the single integral into multiple integrals.

$\int \sec^{2} (x) d x + \int 2 \sin (\frac{x}{4}) d x + \int - \cot (5 a n x) d x$

Step 3

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$\tan (x) + C + \int 2 \sin (\frac{x}{4}) d x + \int - \cot (5 a n x) d x$

Step 4

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

$\tan (x) + C + 2 \int \sin (\frac{x}{4}) d x + \int - \cot (5 a n x) d x$

Step 5

Let $u_{1} = \frac{x}{4}$ . Then $d u_{1} = \frac{1}{4} d x$ , so $4 d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 5.1

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

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Step 5.1.1

Differentiate $\frac{x}{4}$ .

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

Step 5.1.2

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x]$ .

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

Step 5.1.3

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

$\frac{1}{4} \cdot 1$

Step 5.1.4

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

$\frac{1}{4}$

Step 5.2

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

$\tan (x) + C + 2 \int \sin (u_{1}) \frac{1}{\frac{1}{4}} d u_{1} + \int - \cot (5 a n x) d x$

Step 6

Simplify.

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Step 6.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{4}$ .

$\tan (x) + C + 2 \int \sin (u_{1}) (1 \cdot 4) d u_{1} + \int - \cot (5 a n x) d x$

Step 6.2

Multiply $4$ by $1$ .

$\tan (x) + C + 2 \int \sin (u_{1}) \cdot 4 d u_{1} + \int - \cot (5 a n x) d x$

Step 6.3

Move $4$ to the left of $\sin (u_{1})$ .

$\tan (x) + C + 2 \int 4 \sin (u_{1}) d u_{1} + \int - \cot (5 a n x) d x$

Step 7

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

$\tan (x) + C + 2 (4 \int \sin (u_{1}) d u_{1}) + \int - \cot (5 a n x) d x$

Step 8

Multiply $4$ by $2$ .

$\tan (x) + C + 8 \int \sin (u_{1}) d u_{1} + \int - \cot (5 a n x) d x$

Step 9

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

$\tan (x) + C + 8 (- \cos (u_{1}) + C) + \int - \cot (5 a n x) d x$

Step 10

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

$\tan (x) + C + 8 (- \cos (u_{1}) + C) - \int \cot (5 a n x) d x$

Step 11

Let $u_{2} = 5 a n x$ . Then $d u_{2} = 5 a n d x$ , so $\frac{1}{5 a n} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 11.1

Let $u_{2} = 5 a n x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 11.1.1

Differentiate $5 a n x$ .

$\frac{d}{d x} [5 a n x]$

Step 11.1.2

Since $5 a n$ is constant with respect to $x$ , the derivative of $5 a n x$ with respect to $x$ is $5 a n \frac{d}{d x} [x]$ .

$5 a n \frac{d}{d x} [x]$

Step 11.1.3

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

$5 a n \cdot 1$

Step 11.1.4

Multiply $5$ by $1$ .

$5 a n$

Step 11.2

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

$\tan (x) + C + 8 (- \cos (u_{1}) + C) - \int \cot (u_{2}) \frac{1}{5 a n} d u_{2}$

Step 12

Combine $\cot (u_{2})$ and $\frac{1}{5 a n}$ .

$\tan (x) + C + 8 (- \cos (u_{1}) + C) - \int \frac{\cot (u_{2})}{5 a n} d u_{2}$

Step 13

Since $\frac{1}{5 a n}$ is constant with respect to $u_{2}$ , move $\frac{1}{5 a n}$ out of the integral.

$\tan (x) + C + 8 (- \cos (u_{1}) + C) - (\frac{1}{5 a n} \int \cot (u_{2}) d u_{2})$

Step 14

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

$\tan (x) + C + 8 (- \cos (u_{1}) + C) - \frac{1}{5 a n} (\ln (| \sin (u_{2}) |) + C)$

Step 15

Simplify.

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Step 15.1

Simplify.

$\tan (x) - 8 \cos (u_{1}) - \frac{1}{5 a n} \ln (| \sin (u_{2}) |) + C$

Step 15.2

Combine $\ln (| \sin (u_{2}) |)$ and $\frac{1}{5 a n}$ .

$\tan (x) - 8 \cos (u_{1}) - \frac{\ln (| \sin (u_{2}) |)}{5 a n} + C$

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

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

$\tan (x) - 8 \cos (\frac{x}{4}) - \frac{\ln (| \sin (u_{2}) |)}{5 a n} + C$

Step 16.2

Replace all occurrences of $u_{2}$ with $5 a n x$ .

$\tan (x) - 8 \cos (\frac{x}{4}) - \frac{\ln (| \sin (5 a n x) |)}{5 a n} + C$

Step 17

Reorder terms.

$\tan (x) - 8 \cos (\frac{1}{4} x) - \frac{\ln (| \sin (5 a n x) |)}{5 a n} + C$