Calculus Examples

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

Step 1

Let $u = x^{5} - 2$ . Then $d u = 5 x^{4} d x$ , so $\frac{1}{5} d u = x^{4} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = x^{5} - 2$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{5} - 2$ .

$\frac{d}{d x} [x^{5} - 2]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{5} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 2]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 2]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4} + \frac{d}{d x} [- 2]$

Step 1.1.4

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

$5 x^{4} + 0$

Step 1.1.5

Add $5 x^{4}$ and $0$ .

$5 x^{4}$

Step 1.2

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

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

Step 2

Combine $e^{u}$ and $\frac{1}{5}$ .

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

Step 3

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

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

Step 4

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

$\frac{1}{5} (e^{u} + C)$

Step 5

Simplify.

$\frac{1}{5} e^{u} + C$

Step 6

Replace all occurrences of $u$ with $x^{5} - 2$ .

$\frac{1}{5} e^{x^{5} - 2} + C$