Calculus Examples

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

Step 1

Let $u = e^{x}$ . Then $d u = e^{x} d x$ , so $\frac{1}{e^{x}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = e^{x}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $e^{x}$ .

$\frac{d}{d x} [e^{x}]$

Step 1.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x}$

Step 1.2

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

$\int \sin (u) d u$

Step 2

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

$- \cos (u) + C$

Step 3

Replace all occurrences of $u$ with $e^{x}$ .

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