Calculus Examples

$\int \frac{e^{\frac{1}{x^{3}}}}{x^{4}} d x$

Step 1

Let $u = \frac{1}{x^{3}}$ . Then $d u = - \frac{3}{x^{4}} d x$ , so $1 d u = - \frac{3}{x^{4}} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \frac{1}{x^{3}}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\frac{1}{x^{3}}$ .

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

Step 1.1.2

Apply basic rules of exponents.

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Step 1.1.2.1

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

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

Step 1.1.2.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.1.2.2.1

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

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

Step 1.1.2.2.2

Multiply $3$ by $- 1$ .

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

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 3$ .

$- 3 x^{- 4}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

$- 3 \frac{1}{x^{4}}$

Step 1.1.4.2

Combine terms.

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Step 1.1.4.2.1

Combine $- 3$ and $\frac{1}{x^{4}}$ .

$\frac{- 3}{x^{4}}$

Step 1.1.4.2.2

Move the negative in front of the fraction.

$- \frac{3}{x^{4}}$

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 $\frac{1}{x^{3}}$ .

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