Calculus Examples

$\int 9 x^{2} \sqrt{x^{3} - 1} d x$

Step 1

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int x^{2} \sqrt{x^{3} - 1} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $x^{3} - 1$ .

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

Step 2.1.2

By the Sum Rule, the derivative of $x^{3} - 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 1]$ .

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

Step 2.1.3

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

$3 x^{2} + \frac{d}{d x} [- 1]$

Step 2.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$3 x^{2} + 0$

Step 2.1.5

Add $3 x^{2}$ and $0$ .

$3 x^{2}$

Step 2.2

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

$9 \int \sqrt{u} \frac{1}{3} d u$

Step 3

Combine $\sqrt{u}$ and $\frac{1}{3}$ .

$9 \int \frac{\sqrt{u}}{3} d u$

Step 4

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$9 (\frac{1}{3} \int \sqrt{u} d u)$

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $\frac{1}{3}$ and $9$ .

$\frac{9}{3} \int \sqrt{u} d u$

Step 5.1.2

Cancel the common factor of $9$ and $3$ .

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Step 5.1.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 3}{3} \int \sqrt{u} d u$

Step 5.1.2.2

Cancel the common factors.

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Step 5.1.2.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 3}{3 (1)} \int \sqrt{u} d u$

Step 5.1.2.2.2

Cancel the common factor.

$\frac{3 \cdot 3}{3 \cdot 1} \int \sqrt{u} d u$

Step 5.1.2.2.3

Rewrite the expression.

$\frac{3}{1} \int \sqrt{u} d u$

Step 5.1.2.2.4

Divide $3$ by $1$ .

$3 \int \sqrt{u} d u$

Step 5.2

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Rewrite $3 (\frac{2}{3} u^{\frac{3}{2}} + C)$ as $3 (\frac{2}{3}) u^{\frac{3}{2}} + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Combine $3$ and $\frac{2}{3}$ .

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

Step 7.2.2

Multiply $3$ by $2$ .

$\frac{6}{3} u^{\frac{3}{2}} + C$

Step 7.2.3

Cancel the common factor of $6$ and $3$ .

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Step 7.2.3.1

Factor $3$ out of $6$ .

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

Step 7.2.3.2

Cancel the common factors.

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Step 7.2.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 2}{3 (1)} u^{\frac{3}{2}} + C$

Step 7.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 2}{3 \cdot 1} u^{\frac{3}{2}} + C$

Step 7.2.3.2.3

Rewrite the expression.

$\frac{2}{1} u^{\frac{3}{2}} + C$

Step 7.2.3.2.4

Divide $2$ by $1$ .

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

Step 8

Replace all occurrences of $u$ with $x^{3} - 1$ .

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