Calculus Examples

$\int \frac{z}{e^{z}} d z$

Step 1

Simplify the expression.

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Step 1.1

Negate the exponent of $e^{z}$ and move it out of the denominator.

$\int z {(e^{z})}^{- 1} d z$

Step 1.2

Multiply the exponents in ${(e^{z})}^{- 1}$ .

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Step 1.2.1

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

$\int z e^{z \cdot - 1} d z$

Step 1.2.2

Move $- 1$ to the left of $z$ .

$\int z e^{- 1 \cdot z} d z$

Step 1.2.3

Rewrite $- 1 z$ as $- z$ .

$\int z e^{- z} d z$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = z$ and $d v = e^{- z}$ .

$z (- e^{- z}) - \int - e^{- z} d z$

Step 3

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

$z (- e^{- z}) - - \int e^{- z} d z$

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

$z (- e^{- z}) + 1 \int e^{- z} d z$

Step 4.2

Multiply $\int e^{- z} d z$ by $1$ .

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

Step 5

Let $u = - z$ . Then $d u = - d z$ , so $- d u = d z$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = - z$ . Find $\frac{d u}{d z}$ .

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Step 5.1.1

Differentiate $- z$ .

$\frac{d}{d z} [- z]$

Step 5.1.2

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

$- \frac{d}{d z} [z]$

Step 5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d z} [z^{n}]$ is $n z^{n - 1}$ where $n = 1$ .

$- 1 \cdot 1$

Step 5.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 5.2

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

$z (- e^{- z}) + \int - e^{u} d u$

Step 6

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

$z (- e^{- z}) - \int e^{u} d u$

Step 7

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

$z (- e^{- z}) - (e^{u} + C)$

Step 8

Rewrite $z (- e^{- z}) - (e^{u} + C)$ as $- z e^{- z} - e^{u} + C$ .

$- z e^{- z} - e^{u} + C$

Step 9

Replace all occurrences of $u$ with $- z$ .

$- z e^{- z} - e^{- z} + C$