Calculus Examples

$\int \frac{e^{\frac{1}{x}}}{x^{3}} d x$

Step 1

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

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

$- x^{- 2}$

Step 1.1.4

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

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

Step 1.2

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

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{u}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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

Step 7

Replace all occurrences of $u$ with $\frac{1}{x}$ .

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Multiply $- - e^{\frac{1}{x}}$ .

Tap for more steps...

Step 8.3.1

Multiply $- 1$ by $- 1$ .

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

Step 8.3.2

Multiply $e^{\frac{1}{x}}$ by $1$ .

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