Calculus Examples

$\int \sqrt{x^{5} - 8} \cdot 5 x^{4} d x$

Step 1

Move $5$ to the left of $\sqrt{x^{5} - 8}$ .

$\int 5 \sqrt{x^{5} - 8} x^{4} d x$

Step 2

Let $u = x^{5} - 8$ . Then $d u = 5 x^{4} d x$ , so $\frac{1}{5 x^{4}} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = x^{5} - 8$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $x^{5} - 8$ .

$\frac{d}{d x} [x^{5} - 8]$

Step 2.1.2

By the Sum Rule, the derivative of $x^{5} - 8$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8]$ .

$\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 8]$

Step 2.1.3

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

$5 x^{4} + \frac{d}{d x} [- 8]$

Step 2.1.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$5 x^{4} + 0$

Step 2.1.5

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

$5 x^{4}$

Step 2.2

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

$\int \sqrt{u} d u$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\int u^{\frac{1}{2}} d u$

Step 4

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$\frac{2}{3} u^{\frac{3}{2}} + C$

Step 5

Replace all occurrences of $u$ with $x^{5} - 8$ .

$\frac{2}{3} {(x^{5} - 8)}^{\frac{3}{2}} + C$