Calculus Examples

$\int (3 \sec^{2} (4 x) + 3 \csc^{2} (4 x)) d x$

Step 1

Remove parentheses.

$\int 3 \sec^{2} (4 x) + 3 \csc^{2} (4 x) d x$

Step 2

Split the single integral into multiple integrals.

$\int 3 \sec^{2} (4 x) d x + \int 3 \csc^{2} (4 x) d x$

Step 3

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

$3 \int \sec^{2} (4 x) d x + \int 3 \csc^{2} (4 x) d x$

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $4 x$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

$4 \cdot 1$

Step 4.1.4

Multiply $4$ by $1$ .

$4$

Step 4.2

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

$3 \int \sec^{2} (u_{1}) \frac{1}{4} d u_{1} + \int 3 \csc^{2} (4 x) d x$

Step 5

Combine $\sec^{2} (u_{1})$ and $\frac{1}{4}$ .

$3 \int \frac{\sec^{2} (u_{1})}{4} d u_{1} + \int 3 \csc^{2} (4 x) d x$

Step 6

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

$3 (\frac{1}{4} \int \sec^{2} (u_{1}) d u_{1}) + \int 3 \csc^{2} (4 x) d x$

Step 7

Combine $\frac{1}{4}$ and $3$ .

$\frac{3}{4} \int \sec^{2} (u_{1}) d u_{1} + \int 3 \csc^{2} (4 x) d x$

Step 8

Since the derivative of $\tan (u_{1})$ is $\sec^{2} (u_{1})$ , the integral of $\sec^{2} (u_{1})$ is $\tan (u_{1})$ .

$\frac{3}{4} (\tan (u_{1}) + C) + \int 3 \csc^{2} (4 x) d x$

Step 9

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

$\frac{3}{4} (\tan (u_{1}) + C) + 3 \int \csc^{2} (4 x) d x$

Step 10

Let $u_{2} = 4 x$ . Then $d u_{2} = 4 d x$ , so $\frac{1}{4} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 10.1

Let $u_{2} = 4 x$ . Find $\frac{d u_{2}}{d x}$ .

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Step 10.1.1

Differentiate $4 x$ .

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

Step 10.1.2

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

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

Step 10.1.3

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

$4 \cdot 1$

Step 10.1.4

Multiply $4$ by $1$ .

$4$

Step 10.2

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

$\frac{3}{4} (\tan (u_{1}) + C) + 3 \int \csc^{2} (u_{2}) \frac{1}{4} d u_{2}$

Step 11

Combine $\csc^{2} (u_{2})$ and $\frac{1}{4}$ .

$\frac{3}{4} (\tan (u_{1}) + C) + 3 \int \frac{\csc^{2} (u_{2})}{4} d u_{2}$

Step 12

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

$\frac{3}{4} (\tan (u_{1}) + C) + 3 (\frac{1}{4} \int \csc^{2} (u_{2}) d u_{2})$

Step 13

Combine $\frac{1}{4}$ and $3$ .

$\frac{3}{4} (\tan (u_{1}) + C) + \frac{3}{4} \int \csc^{2} (u_{2}) d u_{2}$

Step 14

Since the derivative of $- \cot (u_{2})$ is $\csc^{2} (u_{2})$ , the integral of $\csc^{2} (u_{2})$ is $- \cot (u_{2})$ .

$\frac{3}{4} (\tan (u_{1}) + C) + \frac{3}{4} (- \cot (u_{2}) + C)$

Step 15

Simplify.

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Step 15.1

Simplify.

$\frac{3}{4} \tan (u_{1}) + \frac{3}{4} (- \cot (u_{2})) + C$

Step 15.2

Combine $\frac{3}{4}$ and $\cot (u_{2})$ .

$\frac{3}{4} \tan (u_{1}) - \frac{3 \cot (u_{2})}{4} + C$

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

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

$\frac{3}{4} \tan (4 x) - \frac{3 \cot (u_{2})}{4} + C$

Step 16.2

Replace all occurrences of $u_{2}$ with $4 x$ .

$\frac{3}{4} \tan (4 x) - \frac{3 \cot (4 x)}{4} + C$

Step 17

Reorder terms.

$\frac{3}{4} \tan (4 x) - \frac{3}{4} \cot (4 x) + C$