Calculus Examples

$2 x \sqrt{x}$

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 4.2.2

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 4.2.2.1

Raise $x$ to the power of $1$ .

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

Step 4.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2.3

Write $1$ as a fraction with a common denominator.

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

Add $1$ and $2$ .

$\int 2 x^{\frac{3}{2}} d x$

Step 5

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

$2 \int x^{\frac{3}{2}} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

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

Step 7

Simplify the answer.

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Step 7.1

Rewrite $2 (\frac{2}{5} x^{\frac{5}{2}} + C)$ as $2 (\frac{2}{5}) x^{\frac{5}{2}} + C$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Combine $2$ and $\frac{2}{5}$ .

$\frac{2 \cdot 2}{5} x^{\frac{5}{2}} + C$

Step 7.2.2

Multiply $2$ by $2$ .

$\frac{4}{5} x^{\frac{5}{2}} + C$

Step 8

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

$F (x) =$ $\frac{4}{5} x^{\frac{5}{2}} + C$