Calculus Examples

$\int (3 \sec (6 x) \tan (6 x) - 5 x \csc^{2} (2 x^{2})) d x$

Step 1

Remove parentheses.

$\int 3 \sec (6 x) \tan (6 x) - 5 x \csc^{2} (2 x^{2}) d x$

Step 2

Split the single integral into multiple integrals.

$\int 3 \sec (6 x) \tan (6 x) d x + \int - 5 x \csc^{2} (2 x^{2}) d x$

Step 3

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

$3 \int \sec (6 x) \tan (6 x) d x + \int - 5 x \csc^{2} (2 x^{2}) d x$

Step 4

Let $u_{2} = \sec (6 x)$ . Then $d u_{2} = 6 \sec (6 x) \tan (6 x) d x$ , so $\frac{1}{6} d u_{2} = \sec (6 x) \tan (6 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.1

Let $u_{2} = \sec (6 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 4.1.1

Differentiate $\sec (6 x)$ .

$\frac{d}{d x} [\sec (6 x)]$

Step 4.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = 6 x$ .

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Step 4.1.2.1

To apply the Chain Rule, set $u_{1}$ as $6 x$ .

$\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d x} [6 x]$

Step 4.1.2.2

The derivative of $\sec (u_{1})$ with respect to $u_{1}$ is $\sec (u_{1}) \tan (u_{1})$ .

$\sec (u_{1}) \tan (u_{1}) \frac{d}{d x} [6 x]$

Step 4.1.2.3

Replace all occurrences of $u_{1}$ with $6 x$ .

$\sec (6 x) \tan (6 x) \frac{d}{d x} [6 x]$

Step 4.1.3

Differentiate.

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Step 4.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$\sec (6 x) \tan (6 x) (6 \frac{d}{d x} [x])$

Step 4.1.3.2

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

$\sec (6 x) \tan (6 x) (6 \cdot 1)$

Step 4.1.3.3

Simplify the expression.

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Step 4.1.3.3.1

Multiply $6$ by $1$ .

$\sec (6 x) \tan (6 x) \cdot 6$

Step 4.1.3.3.2

Move $6$ to the left of $\sec (6 x) \tan (6 x)$ .

$6 \sec (6 x) \tan (6 x)$

Step 4.2

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

$3 \int \frac{1}{6} d u_{2} + \int - 5 x \csc^{2} (2 x^{2}) d x$

Step 5

Apply the constant rule.

$3 (\frac{1}{6} u_{2} + C) + \int - 5 x \csc^{2} (2 x^{2}) d x$

Step 6

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$3 (\frac{1}{6} u_{2} + C) - 5 \int x \csc^{2} (2 x^{2}) d x$

Step 7

Let $u_{3} = 2 x^{2}$ . Then $d u_{3} = 4 x d x$ , so $\frac{1}{4} d u_{3} = x d x$ . Rewrite using $u_{3}$ and $d$ $u_{3}$ .

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Step 7.1

Let $u_{3} = 2 x^{2}$ . Find $\frac{d u_{3}}{d x}$ .

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Step 7.1.1

Differentiate $2 x^{2}$ .

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

Step 7.1.2

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

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

Step 7.1.3

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

$2 (2 x)$

Step 7.1.4

Multiply $2$ by $2$ .

$4 x$

Step 7.2

Rewrite the problem using $u_{3}$ and $d u_{3}$ .

$3 (\frac{1}{6} u_{2} + C) - 5 \int \csc^{2} (u_{3}) \frac{1}{4} d u_{3}$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{6}$ and $u_{2}$ .

$3 (\frac{u_{2}}{6} + C) - 5 \int \csc^{2} (u_{3}) \frac{1}{4} d u_{3}$

Step 8.2

Combine $\csc^{2} (u_{3})$ and $\frac{1}{4}$ .

$3 (\frac{u_{2}}{6} + C) - 5 \int \frac{\csc^{2} (u_{3})}{4} d u_{3}$

Step 9

Since $\frac{1}{4}$ is constant with respect to $u_{3}$ , move $\frac{1}{4}$ out of the integral.

$3 (\frac{u_{2}}{6} + C) - 5 (\frac{1}{4} \int \csc^{2} (u_{3}) d u_{3})$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{4}$ and $- 5$ .

$3 (\frac{u_{2}}{6} + C) + \frac{- 5}{4} \int \csc^{2} (u_{3}) d u_{3}$

Step 10.2

Move the negative in front of the fraction.

$3 (\frac{u_{2}}{6} + C) - \frac{5}{4} \int \csc^{2} (u_{3}) d u_{3}$

Step 11

Since the derivative of $- \cot (u_{3})$ is $\csc^{2} (u_{3})$ , the integral of $\csc^{2} (u_{3})$ is $- \cot (u_{3})$ .

$3 (\frac{u_{2}}{6} + C) - \frac{5}{4} (- \cot (u_{3}) + C)$

Step 12

Simplify.

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Step 12.1

Simplify.

$\frac{u_{2}}{2} - \frac{5}{4} (- \cot (u_{3})) + C$

Step 12.2

Simplify.

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Step 12.2.1

Multiply $- 1$ by $- 1$ .

$\frac{u_{2}}{2} + 1 (\frac{5}{4}) \cot (u_{3}) + C$

Step 12.2.2

Multiply $\frac{5}{4}$ by $1$ .

$\frac{u_{2}}{2} + \frac{5}{4} \cot (u_{3}) + C$

Step 12.2.3

Combine $\frac{5}{4}$ and $\cot (u_{3})$ .

$\frac{u_{2}}{2} + \frac{5 \cot (u_{3})}{4} + C$

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

Replace all occurrences of $u_{2}$ with $\sec (6 x)$ .

$\frac{\sec (6 x)}{2} + \frac{5 \cot (u_{3})}{4} + C$

Step 13.2

Replace all occurrences of $u_{3}$ with $2 x^{2}$ .

$\frac{\sec (6 x)}{2} + \frac{5 \cot (2 x^{2})}{4} + C$

Step 14

Reorder terms.

$\frac{1}{2} \sec (6 x) + \frac{5}{4} \cot (2 x^{2}) + C$