Calculus Examples

$\int \sin (2 x) \sec^{5} (2 x) d x$

Step 1

Simplify.

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Step 1.1

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$\int \sin (2 x) {(\frac{1}{\cos (2 x)})}^{5} d x$

Step 1.2

Apply the product rule to $\frac{1}{\cos (2 x)}$ .

$\int \sin (2 x) \frac{1^{5}}{\cos^{5} (2 x)} d x$

Step 1.3

One to any power is one.

$\int \sin (2 x) \frac{1}{\cos^{5} (2 x)} d x$

Step 1.4

Combine $\sin (2 x)$ and $\frac{1}{\cos^{5} (2 x)}$ .

$\int \frac{\sin (2 x)}{\cos^{5} (2 x)} d x$

Step 1.5

Factor $\cos (2 x)$ out of $\cos^{5} (2 x)$ .

$\int \frac{\sin (2 x)}{\cos (2 x) \cos^{4} (2 x)} d x$

Step 1.6

Separate fractions.

$\int \frac{1}{\cos^{4} (2 x)} \cdot \frac{\sin (2 x)}{\cos (2 x)} d x$

Step 1.7

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$\int \frac{1}{\cos^{4} (2 x)} \tan (2 x) d x$

Step 1.8

Combine $\frac{1}{\cos^{4} (2 x)}$ and $\tan (2 x)$ .

$\int \frac{\tan (2 x)}{\cos^{4} (2 x)} d x$

Step 1.9

Factor $\cos (2 x)$ out of $\cos^{4} (2 x)$ .

$\int \frac{\tan (2 x)}{\cos (2 x) \cos^{3} (2 x)} d x$

Step 1.10

Separate fractions.

$\int \frac{1}{\cos^{3} (2 x)} \cdot \frac{\tan (2 x)}{\cos (2 x)} d x$

Step 1.11

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$\int \frac{1}{\cos^{3} (2 x)} \cdot \frac{\frac{\sin (2 x)}{\cos (2 x)}}{\cos (2 x)} d x$

Step 1.12

Rewrite $\frac{\frac{\sin (2 x)}{\cos (2 x)}}{\cos (2 x)}$ as a product.

$\int \frac{1}{\cos^{3} (2 x)} (\frac{\sin (2 x)}{\cos (2 x)} \cdot \frac{1}{\cos (2 x)}) d x$

Step 1.13

Simplify.

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Step 1.13.1

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$\int \frac{1}{\cos^{3} (2 x)} (\tan (2 x) \frac{1}{\cos (2 x)}) d x$

Step 1.13.2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$\int \frac{1}{\cos^{3} (2 x)} (\tan (2 x) \sec (2 x)) d x$

Step 1.14

Multiply $\frac{1}{\cos^{3} (2 x)} (\tan (2 x) \sec (2 x))$ .

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Step 1.14.1

Combine $\tan (2 x)$ and $\frac{1}{\cos^{3} (2 x)}$ .

$\int \frac{\tan (2 x)}{\cos^{3} (2 x)} \sec (2 x) d x$

Step 1.14.2

Combine $\frac{\tan (2 x)}{\cos^{3} (2 x)}$ and $\sec (2 x)$ .

$\int \frac{\tan (2 x) \sec (2 x)}{\cos^{3} (2 x)} d x$

Step 1.15

Factor $\cos (2 x)$ out of $\cos^{3} (2 x)$ .

$\int \frac{\tan (2 x) \sec (2 x)}{\cos (2 x) \cos^{2} (2 x)} d x$

Step 1.16

Separate fractions.

$\int \frac{\sec (2 x)}{\cos^{2} (2 x)} \cdot \frac{\tan (2 x)}{\cos (2 x)} d x$

Step 1.17

Rewrite $\tan (2 x)$ in terms of sines and cosines.

$\int \frac{\sec (2 x)}{\cos^{2} (2 x)} \cdot \frac{\frac{\sin (2 x)}{\cos (2 x)}}{\cos (2 x)} d x$

Step 1.18

Rewrite $\frac{\frac{\sin (2 x)}{\cos (2 x)}}{\cos (2 x)}$ as a product.

$\int \frac{\sec (2 x)}{\cos^{2} (2 x)} (\frac{\sin (2 x)}{\cos (2 x)} \cdot \frac{1}{\cos (2 x)}) d x$

Step 1.19

Simplify.

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Step 1.19.1

Convert from $\frac{\sin (2 x)}{\cos (2 x)}$ to $\tan (2 x)$ .

$\int \frac{\sec (2 x)}{\cos^{2} (2 x)} (\tan (2 x) \frac{1}{\cos (2 x)}) d x$

Step 1.19.2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$\int \frac{\sec (2 x)}{\cos^{2} (2 x)} (\tan (2 x) \sec (2 x)) d x$

Step 1.20

Factor $\cos (2 x)$ out of $\cos^{2} (2 x)$ .

$\int \frac{\sec (2 x)}{\cos (2 x) \cos (2 x)} (\tan (2 x) \sec (2 x)) d x$

Step 1.21

Separate fractions.

$\int \frac{1}{\cos (2 x)} \cdot \frac{\sec (2 x)}{\cos (2 x)} (\tan (2 x) \sec (2 x)) d x$

Step 1.22

Rewrite $\sec (2 x)$ in terms of sines and cosines.

$\int \frac{1}{\cos (2 x)} \cdot \frac{\frac{1}{\cos (2 x)}}{\cos (2 x)} (\tan (2 x) \sec (2 x)) d x$

Step 1.23

Rewrite $\frac{\frac{1}{\cos (2 x)}}{\cos (2 x)}$ as a product.

$\int \frac{1}{\cos (2 x)} (\frac{1}{\cos (2 x)} \cdot \frac{1}{\cos (2 x)}) (\tan (2 x) \sec (2 x)) d x$

Step 1.24

Simplify.

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Step 1.24.1

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$\int \frac{1}{\cos (2 x)} (\sec (2 x) \frac{1}{\cos (2 x)}) (\tan (2 x) \sec (2 x)) d x$

Step 1.24.2

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$\int \frac{1}{\cos (2 x)} (\sec (2 x) \sec (2 x)) (\tan (2 x) \sec (2 x)) d x$

Step 1.24.3

Raise $\sec (2 x)$ to the power of $1$ .

$\int \frac{1}{\cos (2 x)} (\sec^{1} (2 x) \sec (2 x)) (\tan (2 x) \sec (2 x)) d x$

Step 1.24.4

Raise $\sec (2 x)$ to the power of $1$ .

$\int \frac{1}{\cos (2 x)} (\sec^{1} (2 x) \sec^{1} (2 x)) (\tan (2 x) \sec (2 x)) d x$

Step 1.24.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{1}{\cos (2 x)} {\sec (2 x)}^{1 + 1} (\tan (2 x) \sec (2 x)) d x$

Step 1.24.6

Add $1$ and $1$ .

$\int \frac{1}{\cos (2 x)} \sec^{2} (2 x) (\tan (2 x) \sec (2 x)) d x$

Step 1.25

Multiply $\sec^{2} (2 x)$ by $\sec (2 x)$ by adding the exponents.

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Step 1.25.1

Move $\sec (2 x)$ .

$\int \frac{1}{\cos (2 x)} (\sec (2 x) \sec^{2} (2 x)) \tan (2 x) d x$

Step 1.25.2

Multiply $\sec (2 x)$ by $\sec^{2} (2 x)$ .

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Step 1.25.2.1

Raise $\sec (2 x)$ to the power of $1$ .

$\int \frac{1}{\cos (2 x)} (\sec^{1} (2 x) \sec^{2} (2 x)) \tan (2 x) d x$

Step 1.25.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{1}{\cos (2 x)} {\sec (2 x)}^{1 + 2} \tan (2 x) d x$

Step 1.25.3

Add $1$ and $2$ .

$\int \frac{1}{\cos (2 x)} \sec^{3} (2 x) \tan (2 x) d x$

Step 1.26

Convert from $\frac{1}{\cos (2 x)}$ to $\sec (2 x)$ .

$\int \sec (2 x) \sec^{3} (2 x) \tan (2 x) d x$

Step 1.27

Multiply $\sec (2 x)$ by $\sec^{3} (2 x)$ by adding the exponents.

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Step 1.27.1

Multiply $\sec (2 x)$ by $\sec^{3} (2 x)$ .

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Step 1.27.1.1

Raise $\sec (2 x)$ to the power of $1$ .

$\int \sec^{1} (2 x) \sec^{3} (2 x) \tan (2 x) d x$

Step 1.27.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int {\sec (2 x)}^{1 + 3} \tan (2 x) d x$

Step 1.27.2

Add $1$ and $3$ .

$\int \sec^{4} (2 x) \tan (2 x) d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $\sec (2 x)$ .

$\frac{d}{d x} [\sec (2 x)]$

Step 2.1.2

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 $g (x) = 2 x$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\sec (u_{1})] \frac{d}{d x} [2 x]$

Step 2.1.2.2

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

$\sec (u_{1}) \tan (u_{1}) \frac{d}{d x} [2 x]$

Step 2.1.2.3

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

$\sec (2 x) \tan (2 x) \frac{d}{d x} [2 x]$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

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

$\sec (2 x) \tan (2 x) (2 \frac{d}{d x} [x])$

Step 2.1.3.2

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

$\sec (2 x) \tan (2 x) (2 \cdot 1)$

Step 2.1.3.3

Simplify the expression.

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Step 2.1.3.3.1

Multiply $2$ by $1$ .

$\sec (2 x) \tan (2 x) \cdot 2$

Step 2.1.3.3.2

Move $2$ to the left of $\sec (2 x) \tan (2 x)$ .

$2 \sec (2 x) \tan (2 x)$

Step 2.2

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

$\int {u_{2}}^{3} \frac{1}{2} d u_{2}$

Step 3

Combine ${u_{2}}^{3}$ and $\frac{1}{2}$ .

$\int \frac{{u_{2}}^{3}}{2} d u_{2}$

Step 4

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

$\frac{1}{2} \int {u_{2}}^{3} d u_{2}$

Step 5

By the Power Rule, the integral of ${u_{2}}^{3}$ with respect to $u_{2}$ is $\frac{1}{4} {u_{2}}^{4}$ .

$\frac{1}{2} (\frac{1}{4} {u_{2}}^{4} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{2} (\frac{1}{4} {u_{2}}^{4} + C)$ as $\frac{1}{2} \cdot \frac{1}{4} {u_{2}}^{4} + C$ .

$\frac{1}{2} \cdot \frac{1}{4} {u_{2}}^{4} + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$\frac{1}{2 \cdot 4} {u_{2}}^{4} + C$

Step 6.2.2

Multiply $2$ by $4$ .

$\frac{1}{8} {u_{2}}^{4} + C$

Step 7

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

$\frac{1}{8} \sec^{4} (2 x) + C$