Calculus Examples

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

Step 1

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

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Step 1.1

Let $u_{2} = 1 + \sin (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 1.1.1

Differentiate $1 + \sin (2 x)$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 + \sin (2 x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sin (2 x)]$ .

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

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\sin (2 x)]$

Step 1.1.3

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

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Step 1.1.3.1

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

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Step 1.1.3.1.1

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

$0 + \frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [2 x]$

Step 1.1.3.1.2

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

$0 + \cos (u_{1}) \frac{d}{d x} [2 x]$

Step 1.1.3.1.3

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

$0 + \cos (2 x) \frac{d}{d x} [2 x]$

Step 1.1.3.2

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

$0 + \cos (2 x) (2 \frac{d}{d x} [x])$

Step 1.1.3.3

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

$0 + \cos (2 x) (2 \cdot 1)$

Step 1.1.3.4

Multiply $2$ by $1$ .

$0 + \cos (2 x) \cdot 2$

Step 1.1.3.5

Move $2$ to the left of $\cos (2 x)$ .

$0 + 2 \cos (2 x)$

Step 1.1.4

Add $0$ and $2 \cos (2 x)$ .

$2 \cos (2 x)$

Step 1.2

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

$\int \frac{1}{u_{2}} \cdot \frac{1}{2} d u_{2}$

Step 2

Simplify.

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Step 2.1

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

$\int \frac{1}{u_{2} \cdot 2} d u_{2}$

Step 2.2

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

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

Step 3

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

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

Step 4

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

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

Step 5

Simplify.

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

Step 6

Replace all occurrences of $u_{2}$ with $1 + \sin (2 x)$ .

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