Calculus Examples

$\frac{e^{\frac{1}{z}}}{z^{2}} \cdot (d z)$

Step 1

Simplify.

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Step 1.1

Combine $d$ and $\frac{e^{\frac{1}{z}}}{z^{2}}$ .

$\int \frac{d e^{\frac{1}{z}}}{z^{2}} \cdot z d z$

Step 1.2

Combine $\frac{d e^{\frac{1}{z}}}{z^{2}}$ and $z$ .

$\int \frac{d e^{\frac{1}{z}} z}{z^{2}} d z$

Step 1.3

Cancel the common factor of $z$ and $z^{2}$ .

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Step 1.3.1

Factor $z$ out of $d e^{\frac{1}{z}} z$ .

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

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

Factor $z$ out of $z^{2}$ .

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

Step 1.3.2.2

Cancel the common factor.

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

Step 1.3.2.3

Rewrite the expression.

$\int \frac{d e^{\frac{1}{z}}}{z} d z$

Step 2

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

$d \int \frac{e^{\frac{1}{z}}}{z} d z$

Step 3

Let $u = \frac{1}{z}$ . Then $d u = - \frac{1}{z^{2}} d z$ , so $- d u = \frac{1}{z^{2}} d z$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = \frac{1}{z}$ . Find $\frac{d u}{d z}$ .

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Step 3.1.1

Differentiate $\frac{1}{z}$ .

$\frac{d}{d z} [\frac{1}{z}]$

Step 3.1.2

Rewrite $\frac{1}{z}$ as $z^{- 1}$ .

$\frac{d}{d z} [z^{- 1}]$

Step 3.1.3

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

$- z^{- 2}$

Step 3.1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{z^{2}}$

Step 3.2

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

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

Step 4

Simplify.

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Step 4.1

Move ${(\frac{1}{u})}^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

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

Step 4.2

Combine $\frac{1}{u}$ and $e^{u}$ .

$d \int - \frac{e^{u}}{u} d u$

Step 5

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

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

Step 6

$\int \frac{e^{u}}{u} d u$ is a special integral. The integral is the exponential integral function.

$d (- (E i (u) + C))$

Step 7

Simplify.

$d (- E i (u)) + C$