Calculus Examples

$\int \sec (2 x) + \tan (2 x) d x$

Step 1

Split the single integral into multiple integrals.

$\int \sec (2 x) d x + \int \tan (2 x) d x$

Step 2

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

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Step 2.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 2.1.1

Differentiate $2 x$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

The integral of $\sec (u_{1})$ with respect to $u_{1}$ is $\ln (| \sec (u_{1}) + \tan (u_{1}) |)$ .

$\frac{1}{2} (\ln (| \sec (u_{1}) + \tan (u_{1}) |) + C) + \int \tan (2 x) d x$

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $2 x$ .

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

Step 6.1.2

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

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

Step 6.1.3

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

$2 \cdot 1$

Step 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

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

$\frac{1}{2} (\ln (| \sec (u_{1}) + \tan (u_{1}) |) + C) + \int \tan (u_{2}) \frac{1}{2} d u_{2}$

Step 7

Combine $\tan (u_{2})$ and $\frac{1}{2}$ .

$\frac{1}{2} (\ln (| \sec (u_{1}) + \tan (u_{1}) |) + C) + \int \frac{\tan (u_{2})}{2} d u_{2}$

Step 8

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

$\frac{1}{2} (\ln (| \sec (u_{1}) + \tan (u_{1}) |) + C) + \frac{1}{2} \int \tan (u_{2}) d u_{2}$

Step 9

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

$\frac{1}{2} (\ln (| \sec (u_{1}) + \tan (u_{1}) |) + C) + \frac{1}{2} (\ln (| \sec (u_{2}) |) + C)$

Step 10

Simplify.

$\frac{1}{2} \ln (| \sec (u_{1}) + \tan (u_{1}) |) + \frac{1}{2} \ln (| \sec (u_{2}) |) + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

$\frac{1}{2} \ln (| \sec (2 x) + \tan (2 x) |) + \frac{1}{2} \ln (| \sec (u_{2}) |) + C$

Step 11.2

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

$\frac{1}{2} \ln (| \sec (2 x) + \tan (2 x) |) + \frac{1}{2} \ln (| \sec (2 x) |) + C$