Calculus Examples

$\sqrt{x} + 2 x^{2} \sqrt{x}$

Step 1

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

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

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.

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2.4

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

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

Step 4.2.5

Combine the numerators over the common denominator.

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

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $2$ by $2$ .

$\int \sqrt{x} + 2 x^{\frac{1 + 4}{2}} d x$

Step 4.2.6.2

Add $1$ and $4$ .

$\int \sqrt{x} + 2 x^{\frac{5}{2}} d x$

Step 5

Split the single integral into multiple integrals.

$\int \sqrt{x} d x + \int 2 x^{\frac{5}{2}} d x$

Step 6

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

$\int x^{\frac{1}{2}} d x + \int 2 x^{\frac{5}{2}} d x$

Step 7

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

$\frac{2}{3} x^{\frac{3}{2}} + C + \int 2 x^{\frac{5}{2}} d x$

Step 8

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

$\frac{2}{3} x^{\frac{3}{2}} + C + 2 \int x^{\frac{5}{2}} d x$

Step 9

By the Power Rule, the integral of $x^{\frac{5}{2}}$ with respect to $x$ is $\frac{2}{7} x^{\frac{7}{2}}$ .

$\frac{2}{3} x^{\frac{3}{2}} + C + 2 (\frac{2}{7} x^{\frac{7}{2}} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$\frac{2}{3} x^{\frac{3}{2}} + 2 (\frac{2}{7}) x^{\frac{7}{2}} + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $2$ and $\frac{2}{7}$ .

$\frac{2}{3} x^{\frac{3}{2}} + \frac{2 \cdot 2}{7} x^{\frac{7}{2}} + C$

Step 10.2.2

Multiply $2$ by $2$ .

$\frac{2}{3} x^{\frac{3}{2}} + \frac{4}{7} x^{\frac{7}{2}} + C$

Step 11

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

$F (x) =$ $\frac{2}{3} x^{\frac{3}{2}} + \frac{4}{7} x^{\frac{7}{2}} + C$