Calculus Examples

$\int \frac{20 x}{\sqrt[3]{7 - 3 x^{2}}} d x$

Step 1

Since $20$ is constant with respect to $x$ , move $20$ out of the integral.

$20 \int \frac{x}{\sqrt[3]{7 - 3 x^{2}}} d x$

Step 2

Let $u = 7 - 3 x^{2}$ . Then $d u = - 6 x d x$ , so $- \frac{1}{6} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 7 - 3 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $7 - 3 x^{2}$ .

$\frac{d}{d x} [7 - 3 x^{2}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $7 - 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [- 3 x^{2}]$ .

$\frac{d}{d x} [7] + \frac{d}{d x} [- 3 x^{2}]$

Step 2.1.2.2

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 3 x^{2}]$

Step 2.1.3

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

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Step 2.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [x^{2}]$ .

$0 - 3 \frac{d}{d x} [x^{2}]$

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 $n = 2$ .

$0 - 3 (2 x)$

Step 2.1.3.3

Multiply $2$ by $- 3$ .

$0 - 6 x$

Step 2.1.4

Subtract $6 x$ from $0$ .

$- 6 x$

Step 2.2

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

$20 \int \frac{1}{\sqrt[3]{u}} \cdot \frac{1}{- 6} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$20 \int \frac{1}{\sqrt[3]{u}} (- \frac{1}{6}) d u$

Step 3.2

Multiply $\frac{1}{\sqrt[3]{u}}$ by $\frac{1}{6}$ .

$20 \int - \frac{1}{\sqrt[3]{u} \cdot 6} d u$

Step 3.3

Move $6$ to the left of $\sqrt[3]{u}$ .

$20 \int - \frac{1}{6 \sqrt[3]{u}} d u$

Step 4

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

$20 (- \int \frac{1}{6 \sqrt[3]{u}} d u)$

Step 5

Multiply $- 1$ by $20$ .

$- 20 \int \frac{1}{6 \sqrt[3]{u}} d u$

Step 6

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$- 20 (\frac{1}{6} \int \frac{1}{\sqrt[3]{u}} d u)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{1}{6}$ and $- 20$ .

$\frac{- 20}{6} \int \frac{1}{\sqrt[3]{u}} d u$

Step 7.1.2

Cancel the common factor of $- 20$ and $6$ .

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Step 7.1.2.1

Factor $2$ out of $- 20$ .

$\frac{2 (- 10)}{6} \int \frac{1}{\sqrt[3]{u}} d u$

Step 7.1.2.2

Cancel the common factors.

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Step 7.1.2.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot - 10}{2 \cdot 3} \int \frac{1}{\sqrt[3]{u}} d u$

Step 7.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 10}{2 \cdot 3} \int \frac{1}{\sqrt[3]{u}} d u$

Step 7.1.2.2.3

Rewrite the expression.

$\frac{- 10}{3} \int \frac{1}{\sqrt[3]{u}} d u$

Step 7.1.3

Move the negative in front of the fraction.

$- \frac{10}{3} \int \frac{1}{\sqrt[3]{u}} d u$

Step 7.2

Apply basic rules of exponents.

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Step 7.2.1

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

$- \frac{10}{3} \int \frac{1}{u^{\frac{1}{3}}} d u$

Step 7.2.2

Move $u^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{10}{3} \int {(u^{\frac{1}{3}})}^{- 1} d u$

Step 7.2.3

Multiply the exponents in ${(u^{\frac{1}{3}})}^{- 1}$ .

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Step 7.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{10}{3} \int u^{\frac{1}{3} \cdot - 1} d u$

Step 7.2.3.2

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

$- \frac{10}{3} \int u^{\frac{- 1}{3}} d u$

Step 7.2.3.3

Move the negative in front of the fraction.

$- \frac{10}{3} \int u^{- \frac{1}{3}} d u$

Step 8

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

$- \frac{10}{3} (\frac{3}{2} u^{\frac{2}{3}} + C)$

Step 9

Simplify.

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Step 9.1

Rewrite $- \frac{10}{3} (\frac{3}{2} u^{\frac{2}{3}} + C)$ as $- \frac{10}{3} \cdot \frac{3}{2} u^{\frac{2}{3}} + C$ .

$- \frac{10}{3} \cdot \frac{3}{2} u^{\frac{2}{3}} + C$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $\frac{3}{2}$ by $\frac{10}{3}$ .

$- \frac{3 \cdot 10}{2 \cdot 3} u^{\frac{2}{3}} + C$

Step 9.2.2

Multiply $3$ by $10$ .

$- \frac{30}{2 \cdot 3} u^{\frac{2}{3}} + C$

Step 9.2.3

Multiply $2$ by $3$ .

$- \frac{30}{6} u^{\frac{2}{3}} + C$

Step 9.2.4

Cancel the common factor of $30$ and $6$ .

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Step 9.2.4.1

Factor $6$ out of $30$ .

$- \frac{6 \cdot 5}{6} u^{\frac{2}{3}} + C$

Step 9.2.4.2

Cancel the common factors.

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Step 9.2.4.2.1

Factor $6$ out of $6$ .

$- \frac{6 \cdot 5}{6 (1)} u^{\frac{2}{3}} + C$

Step 9.2.4.2.2

Cancel the common factor.

$- \frac{6 \cdot 5}{6 \cdot 1} u^{\frac{2}{3}} + C$

Step 9.2.4.2.3

Rewrite the expression.

$- \frac{5}{1} u^{\frac{2}{3}} + C$

Step 9.2.4.2.4

Divide $5$ by $1$ .

$- 1 \cdot 5 u^{\frac{2}{3}} + C$

Step 9.2.5

Multiply $- 1$ by $5$ .

$- 5 u^{\frac{2}{3}} + C$

Step 10

Replace all occurrences of $u$ with $7 - 3 x^{2}$ .

$- 5 {(7 - 3 x^{2})}^{\frac{2}{3}} + C$