Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $1 - x^{3}$ .

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

Step 1.1.2

Differentiate.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.3

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

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 1.1.3.2

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

$0 - (3 x^{2})$

Step 1.1.3.3

Multiply $3$ by $- 1$ .

$0 - 3 x^{2}$

Step 1.1.4

Subtract $3 x^{2}$ from $0$ .

$- 3 x^{2}$

Step 1.2

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

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

Step 2

Since $- 3$ is constant with respect to $u$ , move $- 3$ out of the integral.

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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 5.2

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

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

Step 6

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

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