Calculus Examples

$\int \frac{24 x}{\sqrt[3]{6 x^{2} + 1}} d x$

Step 1

Let $u = 6 x^{2} + 1$ . Then $d u = 12 x d x$ , so $\frac{1}{12 x} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 6 x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $6 x^{2} + 1$ .

$\frac{d}{d x} [6 x^{2} + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $6 x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [1]$

Step 1.1.3

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 1.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

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 $n = 2$ .

$6 (2 x) + \frac{d}{d x} [1]$

Step 1.1.3.3

Multiply $2$ by $6$ .

$12 x + \frac{d}{d x} [1]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$12 x + 0$

Step 1.1.4.2

Add $12 x$ and $0$ .

$12 x$

Step 1.2

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

$\int \frac{2}{\sqrt[3]{u}} d u$

Step 2

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

Replace all occurrences of $u$ with $6 x^{2} + 1$ .

$3 {(6 x^{2} + 1)}^{\frac{2}{3}} + C$