Calculus Examples

$\sqrt{3 x}$

Step 1

Write $\sqrt{3 x}$ as a function.

$f (x) = \sqrt{3 x}$

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.

$F (x) = \int \sqrt{3 x} d x$

Step 4

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

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

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Step 4.1.1

Differentiate $3 x$ .

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

Step 4.1.2

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

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

Step 4.1.3

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

$3 \cdot 1$

Step 4.1.4

Multiply $3$ by $1$ .

$3$

Step 4.2

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

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

Step 5

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

$\int \frac{\sqrt{u}}{3} d u$

Step 6

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

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

Step 7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Simplify.

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Step 9.2.1

Multiply $\frac{1}{3}$ by $\frac{2}{3}$ .

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

Step 9.2.2

Multiply $3$ by $3$ .

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

Step 10

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

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

Step 11

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

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