Calculus Examples

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

Step 1

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

$\frac{1}{3} \int \frac{e^{\frac{2}{x}}}{x^{2}} d x$

Step 2

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

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Step 2.1

Let $u = \frac{2}{x}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\frac{2}{x}$ .

$\frac{d}{d x} [\frac{2}{x}]$

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $\frac{2}{x}$ with respect to $x$ is $2 \frac{d}{d x} [\frac{1}{x}]$ .

$2 \frac{d}{d x} [\frac{1}{x}]$

Step 2.1.3

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

$2 \frac{d}{d x} [x^{- 1}]$

Step 2.1.4

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

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

Step 2.1.5

Multiply $- 1$ by $2$ .

$- 2 x^{- 2}$

Step 2.1.6

Simplify.

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Step 2.1.6.1

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

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

Step 2.1.6.2

Combine terms.

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Step 2.1.6.2.1

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

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

Step 2.1.6.2.2

Move the negative in front of the fraction.

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

Step 2.2

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

$\frac{1}{3} \int - \frac{1}{2} e^{u} d u$

Step 3

Since $- \frac{1}{2}$ is constant with respect to $u$ , move $- \frac{1}{2}$ out of the integral.

$\frac{1}{3} (- \frac{1}{2} \int e^{u} d u)$

Step 4

Simplify.

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Step 4.1

Multiply $\frac{1}{3}$ by $\frac{1}{2}$ .

$- \frac{1}{3 \cdot 2} \int e^{u} d u$

Step 4.2

Multiply $3$ by $2$ .

$- \frac{1}{6} \int e^{u} d u$

Step 5

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

$- \frac{1}{6} (e^{u} + C)$

Step 6

Simplify.

$- \frac{1}{6} e^{u} + C$

Step 7

Replace all occurrences of $u$ with $\frac{2}{x}$ .

$- \frac{1}{6} e^{\frac{2}{x}} + C$