Calculus Examples

$\int \sin (x) \sin (\cos (x)) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\cos (x)$ .

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

Step 1.1.2

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

$- \sin (x)$

Step 1.2

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

$\int - 1 \sin (u) d u$

Step 2

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \sin (u) d u$

Step 3

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$- (- \cos (u) + C)$

Step 4

Simplify.

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Step 4.1

Simplify.

$- - \cos (u) + C$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $- 1$ by $- 1$ .

$1 \cos (u) + C$

Step 4.2.2

Multiply $\cos (u)$ by $1$ .

$\cos (u) + C$

Step 5

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

$\cos (\cos (x)) + C$