Calculus Examples

$\int 5 x^{3} e^{2 - 3 x^{4}} d x$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int x^{3} e^{2 - 3 x^{4}} d x$

Step 2

Let $u = 2 - 3 x^{4}$ . Then $d u = - 12 x^{3} d x$ , so $- \frac{1}{12} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 2 - 3 x^{4}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $2 - 3 x^{4}$ .

$\frac{d}{d x} [2 - 3 x^{4}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $2 - 3 x^{4}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- 3 x^{4}]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [- 3 x^{4}]$

Step 2.1.2.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 3 x^{4}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 3 x^{4}]$ .

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Step 2.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{4}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{4}]$ .

$0 - 3 \frac{d}{d x} [x^{4}]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$0 - 3 (4 x^{3})$

Step 2.1.3.3

Multiply $4$ by $- 3$ .

$0 - 12 x^{3}$

Step 2.1.4

Subtract $12 x^{3}$ from $0$ .

$- 12 x^{3}$

Step 2.2

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

$5 \int e^{u} \frac{1}{- 12} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$5 \int e^{u} (- \frac{1}{12}) d u$

Step 3.2

Combine $e^{u}$ and $\frac{1}{12}$ .

$5 \int - \frac{e^{u}}{12} d u$

Step 4

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

$5 (- \int \frac{e^{u}}{12} d u)$

Step 5

Multiply $- 1$ by $5$ .

$- 5 \int \frac{e^{u}}{12} d u$

Step 6

Since $\frac{1}{12}$ is constant with respect to $u$ , move $\frac{1}{12}$ out of the integral.

$- 5 (\frac{1}{12} \int e^{u} d u)$

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{12}$ and $- 5$ .

$\frac{- 5}{12} \int e^{u} d u$

Step 7.2

Move the negative in front of the fraction.

$- \frac{5}{12} \int e^{u} d u$

Step 8

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

$- \frac{5}{12} (e^{u} + C)$

Step 9

Simplify.

$- \frac{5}{12} e^{u} + C$

Step 10

Replace all occurrences of $u$ with $2 - 3 x^{4}$ .

$- \frac{5}{12} e^{2 - 3 x^{4}} + C$