Calculus Examples

$\int \frac{e^{\frac{3}{x}}}{x^{2}} d x$

Step 1

Let $u = \frac{3}{x}$ . Then $d u = - \frac{3}{x^{2}} d x$ , so $1 d u = - \frac{3}{x^{2}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{3}{x}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{3}{x}$ .

$\frac{d}{d x} [\frac{3}{x}]$

Step 1.1.2

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{x}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{x}]$ .

$3 \frac{d}{d x} [\frac{1}{x}]$

Step 1.1.3

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

$3 \frac{d}{d x} [x^{- 1}]$

Step 1.1.4

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

$3 (- x^{- 2})$

Step 1.1.5

Multiply $- 1$ by $3$ .

$- 3 x^{- 2}$

Step 1.1.6

Simplify.

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Step 1.1.6.1

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

$- 3 \frac{1}{x^{2}}$

Step 1.1.6.2

Combine terms.

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Step 1.1.6.2.1

Combine $- 3$ and $\frac{1}{x^{2}}$ .

$\frac{- 3}{x^{2}}$

Step 1.1.6.2.2

Move the negative in front of the fraction.

$- \frac{3}{x^{2}}$

Step 1.2

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

$\int - \frac{1}{3} e^{u} d u$

Step 2

Since $- \frac{1}{3}$ is constant with respect to $u$ , move $- \frac{1}{3}$ out of the integral.

$- \frac{1}{3} \int e^{u} d u$

Step 3

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

$- \frac{1}{3} (e^{u} + C)$

Step 4

Simplify.

$- \frac{1}{3} e^{u} + C$

Step 5

Replace all occurrences of $u$ with $\frac{3}{x}$ .

$- \frac{1}{3} e^{\frac{3}{x}} + C$