Calculus Examples

$\int \sqrt[3]{3 - 4 x^{2}} (- 8 x) d x$

Step 1

Since $- 8$ is constant with respect to $x$ , move $- 8$ out of the integral.

$- 8 \int \sqrt[3]{3 - 4 x^{2}} (x) d x$

Step 2

Let $u = 3 - 4 x^{2}$ . Then $d u = - 8 x d x$ , so $- \frac{1}{8} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Differentiate.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 4 x^{2}]$

Step 2.1.3

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

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Step 2.1.3.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x^{2}$ with respect to $x$ is $- 4 \frac{d}{d x} [x^{2}]$ .

$0 - 4 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$0 - 4 (2 x)$

Step 2.1.3.3

Multiply $2$ by $- 4$ .

$0 - 8 x$

Step 2.1.4

Subtract $8 x$ from $0$ .

$- 8 x$

Step 2.2

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

$- 8 \int \sqrt[3]{u} \frac{1}{- 8} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$- 8 \int \sqrt[3]{u} (- \frac{1}{8}) d u$

Step 3.2

Combine $\sqrt[3]{u}$ and $\frac{1}{8}$ .

$- 8 \int - \frac{\sqrt[3]{u}}{8} d u$

Step 4

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

$- 8 (- \int \frac{\sqrt[3]{u}}{8} d u)$

Step 5

Multiply $- 1$ by $- 8$ .

$8 \int \frac{\sqrt[3]{u}}{8} d u$

Step 6

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$8 (\frac{1}{8} \int \sqrt[3]{u} d u)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{1}{8}$ and $8$ .

$\frac{8}{8} \int \sqrt[3]{u} d u$

Step 7.1.2

Cancel the common factor of $8$ .

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Step 7.1.2.1

Cancel the common factor.

$\frac{8}{8} \int \sqrt[3]{u} d u$

Step 7.1.2.2

Rewrite the expression.

$1 \int \sqrt[3]{u} d u$

Step 7.1.3

Multiply $\int \sqrt[3]{u} d u$ by $1$ .

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

Step 7.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{u}$ as $u^{\frac{1}{3}}$ .

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

Step 8

By the Power Rule, the integral of $u^{\frac{1}{3}}$ with respect to $u$ is $\frac{3}{4} u^{\frac{4}{3}}$ .

$\frac{3}{4} u^{\frac{4}{3}} + C$

Step 9

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

$\frac{3}{4} {(3 - 4 x^{2})}^{\frac{4}{3}} + C$