Calculus Examples

$2 \sqrt{x}$

Step 1

Write $2 \sqrt{x}$ as a function.

$f (x) = 2 \sqrt{x}$

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.

$F (x) = \int 2 \sqrt{x} d x$

Step 4

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

$2 \int \sqrt{x} d x$

Step 5

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

$2 \int x^{\frac{1}{2}} d x$

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Multiply $2$ by $2$ .

$\frac{4}{3} x^{\frac{3}{2}} + C$

Step 8

The answer is the antiderivative of the function $f (x) = 2 \sqrt{x}$ .

$F (x) =$ $\frac{4}{3} x^{\frac{3}{2}} + C$