Calculus Examples

$\int \cot (x) \ln (\sin (x)) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\ln (\sin (x))$ .

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

Step 1.1.2

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

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Step 1.1.2.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x)$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.3

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

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

Step 1.1.4

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

$\csc (x) \cos (x)$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Reorder the factors of $\csc (x) \cos (x)$ .

$\cos (x) \csc (x)$

Step 1.1.5.2

Rewrite $\csc (x)$ in terms of sines and cosines.

$\cos (x) \frac{1}{\sin (x)}$

Step 1.1.5.3

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

$\frac{\cos (x)}{\sin (x)}$

Step 1.1.5.4

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

$\cot (x)$

Step 1.2

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

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

Step 2

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

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

Step 3

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

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