Calculus Examples

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

Step 1

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

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Step 1.1

Let $u = x + \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x + \sin (x)$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$1 + \cos (x)$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u} d u$

Step 2

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

$\ln (| u |) + C$

Step 3

Replace all occurrences of $u$ with $x + \sin (x)$ .

$\ln (| x + \sin (x) |) + C$