Calculus Examples

$\int \frac{4 x^{2}}{\sqrt[3]{4 x^{3} + 5}} d x$

Step 1

Let $u = 4 x^{3} + 5$ . Then $d u = 12 x^{2} d x$ , so $\frac{1}{12} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 x^{3} + 5$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x^{3} + 5$ .

$\frac{d}{d x} [4 x^{3} + 5]$

Step 1.1.2

By the Sum Rule, the derivative of $4 x^{3} + 5$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [5]$

Step 1.1.3

Evaluate $\frac{d}{d x} [4 x^{3}]$ .

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Step 1.1.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

$4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [5]$

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 $n = 3$ .

$4 (3 x^{2}) + \frac{d}{d x} [5]$

Step 1.1.3.3

Multiply $3$ by $4$ .

$12 x^{2} + \frac{d}{d x} [5]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$12 x^{2} + 0$

Step 1.1.4.2

Add $12 x^{2}$ and $0$ .

$12 x^{2}$

Step 1.2

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

$\int \frac{4}{\sqrt[3]{u}} \cdot \frac{1}{12} d u$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{4}{\sqrt[3]{u}}$ by $\frac{1}{12}$ .

$\int \frac{4}{\sqrt[3]{u} \cdot 12} d u$

Step 2.2

Move $12$ to the left of $\sqrt[3]{u}$ .

$\int \frac{4}{12 \cdot \sqrt[3]{u}} d u$

Step 2.3

Cancel the common factor of $4$ and $12$ .

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Step 2.3.1

Factor $4$ out of $4$ .

$\int \frac{4 (1)}{12 \sqrt[3]{u}} d u$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $4$ out of $12 \sqrt[3]{u}$ .

$\int \frac{4 (1)}{4 (3 \sqrt[3]{u})} d u$

Step 2.3.2.2

Cancel the common factor.

$\int \frac{4 \cdot 1}{4 (3 \sqrt[3]{u})} d u$

Step 2.3.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Replace all occurrences of $u$ with $4 x^{3} + 5$ .

$\frac{1}{2} {(4 x^{3} + 5)}^{\frac{2}{3}} + C$