Calculus Examples

$\int \frac{e^{x}}{{(1 - e^{x})}^{2}} d x$

Step 1

Let $u = 1 - e^{x}$ . 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 $u = 1 - e^{x}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 - e^{x}$ .

$\frac{d}{d x} [1 - e^{x}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 - e^{x}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- e^{x}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- e^{x}]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

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

Step 1.1.3

Evaluate $\frac{d}{d x} [- e^{x}]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- e^{x}$ with respect to $x$ is $- \frac{d}{d x} [e^{x}]$ .

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

Step 1.1.3.2

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

Subtract $e^{x}$ from $0$ .

$- e^{x}$

Step 1.2

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

$\int \frac{- 1}{u^{2}} d u$

Step 2

Move the negative in front of the fraction.

$\int - \frac{1}{u^{2}} d u$

Step 3

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

$- \int \frac{1}{u^{2}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

$- \int {(u^{2})}^{- 1} d u$

Step 4.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 4.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \int u^{2 \cdot - 1} d u$

Step 4.2.2

Multiply $2$ by $- 1$ .

$- \int u^{- 2} d u$

Step 5

By the Power Rule, the integral of $u^{- 2}$ with respect to $u$ is $- u^{- 1}$ .

$- (- u^{- 1} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $- (- u^{- 1} + C)$ as $- - \frac{1}{u} + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

Multiply $- 1$ by $- 1$ .

$1 \frac{1}{u} + C$

Step 6.2.2

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

$\frac{1}{u} + C$

Step 7

Replace all occurrences of $u$ with $1 - e^{x}$ .

$\frac{1}{1 - e^{x}} + C$