Calculus Examples

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

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

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

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

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

Set up the integral to solve.

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

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

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Differentiate $\sin (x)$ .

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

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

$\cos (x)$

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

$\int e^{u} d u$

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

$e^{u} + C$

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

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

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

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