Evaluate integral of 1/(sec(x)^2) with respect to x

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

Simplify.

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Rewrite $1$ as $1^{2}$ .

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

Rewrite $\frac{1^{2}}{\sec^{2} (x)}$ as ${(\frac{1}{\sec (x)})}^{2}$ .

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

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

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

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Multiply $\cos (x)$ by $1$ .

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

Use the half–angle formula to rewrite $\cos^{2} (x)$ as $\frac{1 + \cos (2 x)}{2}$ .

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

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

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

Split the single integral into multiple integrals.

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

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

$\frac{1}{2} (x + C + \int \cos (2 x) d x)$

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

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

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Rewrite.

$\frac{2}{1}$

Divide $2$ by $1$ .

$2$

Rewrite the problem using $u$ and $d u$ .

$\frac{1}{2} (x + C + \int \cos (u) \frac{1}{2} d u)$

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{1}{2} (x + C + \int \frac{\cos (u)}{2} d u)$

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

$\frac{1}{2} (x + C + \frac{1}{2} \int \cos (u) d u)$

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{1}{2} (x + C + \frac{1}{2} (\sin (u) + C))$

Simplify.

$\frac{1}{2} (x + \frac{1}{2} \sin (u)) + C$

Replace all occurrences of $u$ with $2 x$ .

$\frac{1}{2} (x + \frac{1}{2} \sin (2 x)) + C$

Simplify the answer.

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

$\frac{1}{2} (x + \frac{\sin (2 x)}{2}) + C$

Apply the distributive property.

$\frac{1}{2} x + \frac{1}{2} \cdot \frac{\sin (2 x)}{2} + C$

Combine $\frac{1}{2}$ and $x$ .

$\frac{x}{2} + \frac{1}{2} \cdot \frac{\sin (2 x)}{2} + C$

Multiply $\frac{1}{2} \cdot \frac{\sin (2 x)}{2}$ .

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Multiply $\frac{1}{2}$ and $\frac{\sin (2 x)}{2}$ .

$\frac{x}{2} + \frac{\sin (2 x)}{2 \cdot 2} + C$

Multiply $2$ by $2$ .

$\frac{x}{2} + \frac{\sin (2 x)}{4} + C$

Reorder terms.

$\frac{1}{2} x + \frac{1}{4} \sin (2 x) + C$

Evaluate integral of 1/(sec(x)^2) with respect to x

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