Calculus Examples

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

Step 1

Use the double-angle identity to transform $\cos (x)$ to $\cos^{2} (\frac{x}{2}) - \sin^{2} (\frac{x}{2})$ .

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

Step 2

Apply the distributive property.

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

Step 3

Multiply $- 1$ by $2$ .

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

Step 4

Use the pythagorean identity to transform $3$ to $3 \sin^{2} (\frac{x}{2}) + 3 \cos^{2} (\frac{x}{2})$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Subtract $2 {\sin (\frac{x}{2})}^{2}$ from $3 {\sin (\frac{x}{2})}^{2}$ .

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

Step 5.2

Add $3 {\cos (\frac{x}{2})}^{2}$ and $2 {\cos (\frac{x}{2})}^{2}$ .

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

Step 6

Multiply the argument by $\frac{\sec^{2} (\frac{x}{2})}{\sec^{2} (\frac{x}{2})}$

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

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

Combine.

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

Step 7.2

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

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

Step 7.3

Apply the distributive property.

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

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 8.3

One to any power is one.

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

Step 8.4

Cancel the common factor of $\cos^{2} (\frac{x}{2})$ .

Tap for more steps...

Step 8.4.1

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

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

Step 8.4.2

Cancel the common factor.

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

Step 8.4.3

Rewrite the expression.

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

Step 8.5

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

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

Step 8.6

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

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

Step 8.7

One to any power is one.

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

Step 8.8

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

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

Step 9

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

$\int \frac{\sec^{2} (\frac{x}{2})}{5 + \tan^{2} (\frac{x}{2})} d x$

Step 10

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

Tap for more steps...

Step 10.1

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

Tap for more steps...

Step 10.1.1

Differentiate $\frac{x}{2}$ .

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

Step 10.1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{1}{2} \frac{d}{d x} [x]$

Step 10.1.3

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

$\frac{1}{2} \cdot 1$

Step 10.1.4

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

$\frac{1}{2}$

Step 10.2

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

$\int \frac{\sec^{2} (u_{1})}{5 + \tan^{2} (u_{1})} \cdot \frac{1}{\frac{1}{2}} d u_{1}$

Step 11

Simplify.

Tap for more steps...

Step 11.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\int \frac{\sec^{2} (u_{1})}{5 + \tan^{2} (u_{1})} (1 \cdot 2) d u_{1}$

Step 11.2

Multiply $2$ by $1$ .

$\int \frac{\sec^{2} (u_{1})}{5 + \tan^{2} (u_{1})} \cdot 2 d u_{1}$

Step 11.3

Combine $\frac{\sec^{2} (u_{1})}{5 + \tan^{2} (u_{1})}$ and $2$ .

$\int \frac{\sec^{2} (u_{1}) \cdot 2}{5 + \tan^{2} (u_{1})} d u_{1}$

Step 11.4

Move $2$ to the left of $\sec^{2} (u_{1})$ .

$\int \frac{2 \sec^{2} (u_{1})}{5 + \tan^{2} (u_{1})} d u_{1}$

Step 12

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

$2 \int \frac{\sec^{2} (u_{1})}{5 + \tan^{2} (u_{1})} d u_{1}$

Step 13

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

Tap for more steps...

Step 13.1

Let $u_{2} = \tan (u_{1})$ . Find $\frac{d u_{2}}{d u_{1}}$ .

Tap for more steps...

Step 13.1.1

Differentiate $\tan (u_{1})$ .

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

Step 13.1.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$\sec^{2} (u_{1})$

Step 13.2

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

$2 \int \frac{1}{5 + {u_{2}}^{2}} d u_{2}$

Step 14

Rewrite $5$ as ${\sqrt{5}}^{2}$ .

$2 \int \frac{1}{{\sqrt{5}}^{2} + {u_{2}}^{2}} d u_{2}$

Step 15

The integral of $\frac{1}{{\sqrt{5}}^{2} + {u_{2}}^{2}}$ with respect to $u_{2}$ is $\frac{1}{\sqrt{5}} \arctan (\frac{u_{2}}{\sqrt{5}}) + C$ .

$2 (\frac{1}{\sqrt{5}} \arctan (\frac{u_{2}}{\sqrt{5}}) + C)$

Step 16

Simplify.

Tap for more steps...

Step 16.1

Combine $\frac{1}{\sqrt{5}}$ and $\arctan (\frac{u_{2}}{\sqrt{5}})$ .

$2 (\frac{\arctan (\frac{u_{2}}{\sqrt{5}})}{\sqrt{5}} + C)$

Step 16.2

Rewrite $2 (\frac{\arctan (\frac{u_{2}}{\sqrt{5}})}{\sqrt{5}} + C)$ as $2 \frac{\arctan (\frac{u_{2} \sqrt{5}}{5})}{\sqrt{5}} + C$ .

$2 \frac{\arctan (\frac{u_{2} \sqrt{5}}{5})}{\sqrt{5}} + C$

Step 16.3

Combine $2$ and $\frac{\arctan (\frac{u_{2} \sqrt{5}}{5})}{\sqrt{5}}$ .

$\frac{2 \arctan (\frac{u_{2} \sqrt{5}}{5})}{\sqrt{5}} + C$

Step 17

Substitute back in for each integration substitution variable.

Tap for more steps...

Step 17.1

Replace all occurrences of $u_{2}$ with $\tan (u_{1})$ .

$\frac{2 \arctan (\frac{\tan (u_{1}) \sqrt{5}}{5})}{\sqrt{5}} + C$

Step 17.2

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

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5})}{\sqrt{5}} + C$

Step 18

Simplify.

Tap for more steps...

Step 18.1

Multiply $\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5})}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5})}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} + C$

Step 18.2

Combine and simplify the denominator.

Tap for more steps...

Step 18.2.1

Multiply $\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5})}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{\sqrt{5} \sqrt{5}} + C$

Step 18.2.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} + C$

Step 18.2.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} + C$

Step 18.2.4

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

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{{\sqrt{5}}^{1 + 1}} + C$

Step 18.2.5

Add $1$ and $1$ .

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{{\sqrt{5}}^{2}} + C$

Step 18.2.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Step 18.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} + C$

Step 18.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} + C$

Step 18.2.6.3

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

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{5^{\frac{2}{2}}} + C$

Step 18.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 18.2.6.4.1

Cancel the common factor.

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{5^{\frac{2}{2}}} + C$

Step 18.2.6.4.2

Rewrite the expression.

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{5^{1}} + C$

Step 18.2.6.5

Evaluate the exponent.

$\frac{2 \arctan (\frac{\tan (\frac{x}{2}) \sqrt{5}}{5}) \sqrt{5}}{5} + C$

Step 18.3

Reorder terms.

$\frac{2 \sqrt{5}}{5} \arctan (\frac{\sqrt{5}}{5} \tan (\frac{1}{2} x)) + C$