Calculus Examples

$\frac{3}{\sqrt[3]{x + 3}}$

Step 1

Write $\frac{3}{\sqrt[3]{x + 3}}$ as a function.

$f (x) = \frac{3}{\sqrt[3]{x + 3}}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{3}{\sqrt[3]{x + 3}} d x$

Step 4

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

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

Step 5

Let $u = x + 3$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

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

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Step 5.1.1

Differentiate $x + 3$ .

$\frac{d}{d x} [x + 3]$

Step 5.1.2

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

$\frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 5.1.3

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

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

Step 5.1.4

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

$1 + 0$

Step 5.1.5

Add $1$ and $0$ .

$1$

Step 5.2

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

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

Step 6

Apply basic rules of exponents.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

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

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

Step 6.3.3

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Simplify.

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Step 8.2.1

Combine $3$ and $\frac{3}{2}$ .

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

Step 8.2.2

Multiply $3$ by $3$ .

$\frac{9}{2} u^{\frac{2}{3}} + C$

Step 9

Replace all occurrences of $u$ with $x + 3$ .

$\frac{9}{2} {(x + 3)}^{\frac{2}{3}} + C$

Step 10

The answer is the antiderivative of the function $f (x) = \frac{3}{\sqrt[3]{x + 3}}$ .

$F (x) =$ $\frac{9}{2} {(x + 3)}^{\frac{2}{3}} + C$