Calculus Examples

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

Step 1

Since $\frac{2}{3}$ is constant with respect to $x$ , move $\frac{2}{3}$ out of the integral.

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

Step 2

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

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Step 2.1

Let $u = \frac{1}{x^{2}}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\frac{1}{x^{2}}$ .

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

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 2.1.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 2$ .

$- 2 x^{- 3}$

Step 2.1.4

Simplify.

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Step 2.1.4.1

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

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

Step 2.1.4.2

Combine terms.

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Step 2.1.4.2.1

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

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

Step 2.1.4.2.2

Move the negative in front of the fraction.

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

Step 2.2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Multiply $3$ by $2$ .

$- \frac{2}{6} \int e^{u} d u$

Step 4.3

Cancel the common factor of $2$ and $6$ .

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Step 4.3.1

Factor $2$ out of $2$ .

$- \frac{2 (1)}{6} \int e^{u} d u$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $6$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 5

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

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

Step 6

Simplify.

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

Step 7

Replace all occurrences of $u$ with $\frac{1}{x^{2}}$ .

$- \frac{1}{3} e^{\frac{1}{x^{2}}} + C$