Calculus Examples

$\int {(4 + e^{x})}^{- 4} e^{x} d x$

Step 1

Let $u = 4 + e^{x}$ . Then $d u = e^{x} d x$ , so $\frac{1}{e^{x}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 4 + e^{x}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $4 + e^{x}$ .

$\frac{d}{d x} [4 + e^{x}]$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $4 + e^{x}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [e^{x}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [e^{x}]$

Step 1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [e^{x}]$

Step 1.1.3

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

$0 + e^{x}$

Step 1.1.4

Add $0$ and $e^{x}$ .

$e^{x}$

Step 1.2

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

$\int u^{- 4} d u$

Step 2

By the Power Rule, the integral of $u^{- 4}$ with respect to $u$ is $- \frac{1}{3} u^{- 3}$ .

$- \frac{1}{3} u^{- 3} + C$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Rewrite $- \frac{1}{3} u^{- 3} + C$ as $- \frac{1}{3} \cdot \frac{1}{u^{3}} + C$ .

$- \frac{1}{3} \cdot \frac{1}{u^{3}} + C$

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Multiply $\frac{1}{u^{3}}$ by $\frac{1}{3}$ .

$- \frac{1}{u^{3} \cdot 3} + C$

Step 3.2.2

Move $3$ to the left of $u^{3}$ .

$- \frac{1}{3 u^{3}} + C$

Step 4

Replace all occurrences of $u$ with $4 + e^{x}$ .

$- \frac{1}{3 {(4 + e^{x})}^{3}} + C$