Calculus Examples

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

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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

Step 2

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

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

Step 3

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

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

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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Step 6.1

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

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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 $u$ and $d u$ .

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

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

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

Step 12.4

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

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Step 12.4.1

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

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

Step 12.4.2

Multiply $2$ by $2$ .

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

Step 13

Reorder terms.

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