Calculus Examples

$\int \frac{1}{1 - \sin (x)} d x$

Step 1

Apply the sine double-angle identity.

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

Step 2

Multiply $2$ by $- 1$ .

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

Step 3

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

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

Step 4

Combine fractions.

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Step 4.1

Combine.

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

Step 4.2

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

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

Step 5

Simplify denominator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

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

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

Step 6

Simplify each term.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

One to any power is one.

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

Step 6.4

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

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

Step 6.5

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

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

Step 6.6

One to any power is one.

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

Step 6.7

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

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Step 6.7.1

Factor $\cos (\frac{x}{2})$ out of $- 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})$ .

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

Step 6.7.2

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

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

Step 6.7.3

Cancel the common factor.

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

Step 6.7.4

Rewrite the expression.

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

Step 6.8

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

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

Step 6.9

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

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

Step 6.10

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

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

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

Step 9

Transform $\sec^{2} (x)$ to $1 + \tan^{2} (x)$ .

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

Step 10

Multiply $2$ by $- 1$ .

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

Step 11

Factor using the perfect square rule.

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Step 11.1

Rearrange terms.

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

Step 11.2

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

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

Step 11.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 \tan (\frac{x}{2}) = 2 \cdot 1 \cdot \tan (\frac{x}{2})$

Step 11.4

Rewrite the polynomial.

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

Step 11.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = \tan (\frac{x}{2})$ .

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

Step 12

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 $d$ $u_{1}$ .

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Step 12.1

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

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Step 12.1.1

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

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

Step 12.1.2

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 12.1.3

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 12.1.4

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

$\frac{1}{2}$

Step 12.2

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

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

Step 13

Simplify.

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Step 13.1

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

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

Step 13.2

Multiply $2$ by $1$ .

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

Step 13.3

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

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

Step 13.4

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

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

Step 14

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

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

Step 15

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

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Step 15.1

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

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Step 15.1.1

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

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

Step 15.1.2

Differentiate.

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Step 15.1.2.1

By the Sum Rule, the derivative of $1 - \tan (u_{1})$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [1] + \frac{d}{d u_{1}} [- \tan (u_{1})]$ .

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

Step 15.1.2.2

Since $1$ is constant with respect to $u_{1}$ , the derivative of $1$ with respect to $u_{1}$ is $0$ .

$0 + \frac{d}{d u_{1}} [- \tan (u_{1})]$

Step 15.1.3

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

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Step 15.1.3.1

Since $- 1$ is constant with respect to $u_{1}$ , the derivative of $- \tan (u_{1})$ with respect to $u_{1}$ is $- \frac{d}{d u_{1}} [\tan (u_{1})]$ .

$0 - \frac{d}{d u_{1}} [\tan (u_{1})]$

Step 15.1.3.2

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

$0 - \sec^{2} (u_{1})$

Step 15.1.4

Subtract $\sec^{2} (u_{1})$ from $0$ .

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

Step 15.2

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

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

Step 16

Move the negative in front of the fraction.

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

Step 17

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

$2 (- \int \frac{1}{{u_{2}}^{2}} d u_{2})$

Step 18

Simplify the expression.

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Step 18.1

Multiply $- 1$ by $2$ .

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

Step 18.2

Move ${u_{2}}^{2}$ out of the denominator by raising it to the $- 1$ power.

$- 2 \int {({u_{2}}^{2})}^{- 1} d u_{2}$

Step 18.3

Multiply the exponents in ${({u_{2}}^{2})}^{- 1}$ .

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Step 18.3.1

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

$- 2 \int {u_{2}}^{2 \cdot - 1} d u_{2}$

Step 18.3.2

Multiply $2$ by $- 1$ .

$- 2 \int {u_{2}}^{- 2} d u_{2}$

Step 19

By the Power Rule, the integral of ${u_{2}}^{- 2}$ with respect to $u_{2}$ is $- {u_{2}}^{- 1}$ .

$- 2 (- {u_{2}}^{- 1} + C)$

Step 20

Simplify.

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Step 20.1

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

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

Step 20.2

Simplify.

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Step 20.2.1

Multiply $- 1$ by $- 2$ .

$2 \frac{1}{u_{2}} + C$

Step 20.2.2

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

$\frac{2}{u_{2}} + C$

Step 21

Substitute back in for each integration substitution variable.

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Step 21.1

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

$\frac{2}{1 - \tan (u_{1})} + C$

Step 21.2

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

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

Step 22

Reorder terms.

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