Calculus Examples

$\cos (x) \cdot e^{\sin (x)}$

Step 1

Write $\cos (x) \cdot e^{\sin (x)}$ as a function.

$f (x) = \cos (x) \cdot e^{\sin (x)}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

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

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Step 4.1

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

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Step 4.1.1

Differentiate $\sin (x)$ .

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

Step 4.1.2

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

$\cos (x)$

Step 4.2

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

$\int e^{u} d u$

Step 5

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

$e^{u} + C$

Step 6

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

$e^{\sin (x)} + C$

Step 7

The answer is the antiderivative of the function $f (x) = \cos (x) \cdot e^{\sin (x)}$ .

$F (x) =$ $e^{\sin (x)} + C$