Calculus Examples

$\int \frac{2 \sin (2 x)}{1 + 9 \cos^{2} (2 x)} d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \frac{\sin (2 x)}{1 + 9 \cos^{2} (2 x)} d x$

Step 2

Let $u_{2} = \cos (2 x)$ . Then $d u_{2} = - 2 \sin (2 x) d x$ , so $- \frac{1}{2} d u_{2} = \sin (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

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

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Step 2.1.1

Differentiate $\cos (2 x)$ .

$\frac{d}{d x} [\cos (2 x)]$

Step 2.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 2 x$ .

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Step 2.1.2.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$\frac{d}{d u_{1}} [\cos (u_{1})] \frac{d}{d x} [2 x]$

Step 2.1.2.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

$- \sin (u_{1}) \frac{d}{d x} [2 x]$

Step 2.1.2.3

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

$- \sin (2 x) \frac{d}{d x} [2 x]$

Step 2.1.3

Differentiate.

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Step 2.1.3.1

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

$- \sin (2 x) (2 \frac{d}{d x} [x])$

Step 2.1.3.2

Multiply $2$ by $- 1$ .

$- 2 \sin (2 x) \frac{d}{d x} [x]$

Step 2.1.3.3

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

$- 2 \sin (2 x) \cdot 1$

Step 2.1.3.4

Multiply $- 2$ by $1$ .

$- 2 \sin (2 x)$

Step 2.2

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

$2 \int \frac{1}{1 + 9 {u_{2}}^{2}} \cdot \frac{1}{- 2} d u_{2}$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$2 \int \frac{1}{1 + 9 {u_{2}}^{2}} (- \frac{1}{2}) d u_{2}$

Step 3.2

Multiply $\frac{1}{1 + 9 {u_{2}}^{2}}$ by $\frac{1}{2}$ .

$2 \int - \frac{1}{(1 + 9 {u_{2}}^{2}) \cdot 2} d u_{2}$

Step 3.3

Move $2$ to the left of $1 + 9 {u_{2}}^{2}$ .

$2 \int - \frac{1}{2 (1 + 9 {u_{2}}^{2})} d u_{2}$

Step 4

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

$2 (- \int \frac{1}{2 (1 + 9 {u_{2}}^{2})} d u_{2})$

Step 5

Multiply $- 1$ by $2$ .

$- 2 \int \frac{1}{2 (1 + 9 {u_{2}}^{2})} d u_{2}$

Step 6

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

$- 2 (\frac{1}{2} \int \frac{1}{1 + 9 {u_{2}}^{2}} d u_{2})$

Step 7

Simplify.

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Step 7.1

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

$\frac{- 2}{2} \int \frac{1}{1 + 9 {u_{2}}^{2}} d u_{2}$

Step 7.2

Cancel the common factor of $- 2$ and $2$ .

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Step 7.2.1

Factor $2$ out of $- 2$ .

$\frac{2 \cdot - 1}{2} \int \frac{1}{1 + 9 {u_{2}}^{2}} d u_{2}$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 1}{2 (1)} \int \frac{1}{1 + 9 {u_{2}}^{2}} d u_{2}$

Step 7.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 1}{2 \cdot 1} \int \frac{1}{1 + 9 {u_{2}}^{2}} d u_{2}$

Step 7.2.2.3

Rewrite the expression.

$\frac{- 1}{1} \int \frac{1}{1 + 9 {u_{2}}^{2}} d u_{2}$

Step 7.2.2.4

Divide $- 1$ by $1$ .

$- \int \frac{1}{1 + 9 {u_{2}}^{2}} d u_{2}$

Step 8

Factor $9$ out of $1 + 9 {u_{2}}^{2}$ .

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Step 8.1

Factor $9$ out of $1$ .

$- \int \frac{1}{9 (\frac{1}{9}) + 9 {u_{2}}^{2}} d u_{2}$

Step 8.2

Factor $9$ out of $9 {u_{2}}^{2}$ .

$- \int \frac{1}{9 (\frac{1}{9}) + 9 ({u_{2}}^{2})} d u_{2}$

Step 8.3

Factor $9$ out of $9 (\frac{1}{9}) + 9 ({u_{2}}^{2})$ .

$- \int \frac{1}{9 (\frac{1}{9} + {u_{2}}^{2})} d u_{2}$

Step 9

Since $\frac{1}{9}$ is constant with respect to $u_{2}$ , move $\frac{1}{9}$ out of the integral.

$- (\frac{1}{9} \int \frac{1}{\frac{1}{9} + {u_{2}}^{2}} d u_{2})$

Step 10

Rewrite $\frac{1}{9}$ as ${(\frac{1}{3})}^{2}$ .

$- \frac{1}{9} \int \frac{1}{{(\frac{1}{3})}^{2} + {u_{2}}^{2}} d u_{2}$

Step 11

The integral of $\frac{1}{{(\frac{1}{3})}^{2} + {u_{2}}^{2}}$ with respect to $u_{2}$ is $\frac{1}{\frac{1}{3}} \arctan (\frac{u_{2}}{\frac{1}{3}}) + C$ .

$- \frac{1}{9} (\frac{1}{\frac{1}{3}} \arctan (\frac{u_{2}}{\frac{1}{3}}) + C)$

Step 12

Simplify.

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Step 12.1

Simplify.

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Step 12.1.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$- \frac{1}{9} (1 \cdot 3 \arctan (\frac{u_{2}}{\frac{1}{3}}) + C)$

Step 12.1.2

Multiply $3$ by $1$ .

$- \frac{1}{9} (3 \arctan (\frac{u_{2}}{\frac{1}{3}}) + C)$

Step 12.1.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$- \frac{1}{9} (3 \arctan (u_{2} \cdot 3) + C)$

Step 12.1.4

Move $3$ to the left of $u_{2}$ .

$- \frac{1}{9} (3 \arctan (3 u_{2}) + C)$

Step 12.2

Rewrite $- \frac{1}{9} (3 \arctan (3 u_{2}) + C)$ as $- \frac{1}{9} \cdot 3 \arctan (3 u_{2}) + C$ .

$- \frac{1}{9} \cdot 3 \arctan (3 u_{2}) + C$

Step 12.3

Simplify.

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Step 12.3.1

Multiply $3$ by $- 1$ .

$- 3 (\frac{1}{9}) \arctan (3 u_{2}) + C$

Step 12.3.2

Combine $- 3$ and $\frac{1}{9}$ .

$\frac{- 3}{9} \arctan (3 u_{2}) + C$

Step 12.3.3

Cancel the common factor of $- 3$ and $9$ .

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Step 12.3.3.1

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{9} \arctan (3 u_{2}) + C$

Step 12.3.3.2

Cancel the common factors.

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Step 12.3.3.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot - 1}{3 \cdot 3} \arctan (3 u_{2}) + C$

Step 12.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot - 1}{3 \cdot 3} \arctan (3 u_{2}) + C$

Step 12.3.3.2.3

Rewrite the expression.

$\frac{- 1}{3} \arctan (3 u_{2}) + C$

Step 12.3.4

Move the negative in front of the fraction.

$- \frac{1}{3} \arctan (3 u_{2}) + C$

Step 13

Replace all occurrences of $u_{2}$ with $\cos (2 x)$ .

$- \frac{1}{3} \arctan (3 \cos (2 x)) + C$