Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $1 - e^{u}$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

$0 - e^{u}$

Step 1.1.4

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

$- e^{u}$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

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

Step 2

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $- 1$ .

$- \int {u_{1}}^{- 2} d u_{1}$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

Multiply $- 1$ by $- 1$ .

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

Step 6.2.2

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

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

Step 7

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

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