Calculus Examples

$\int (e^{\cos (x)}) (\sin (x)) d x$

Step 1

Multiply $e^{\cos (x)}$ by $\sin (x)$ .

$\int e^{\cos (x)} \sin (x) d x$

Step 2

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 2.1

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

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Step 2.1.1

Differentiate $\cos (x)$ .

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

Step 2.1.2

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

$- \sin (x)$

Step 2.2

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

$\int - e^{u} d u$

Step 3

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

$- \int e^{u} d u$

Step 4

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$- (e^{u} + C)$

Step 5

Simplify.

$- e^{u} + C$

Step 6

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

$- e^{\cos (x)} + C$