Calculus Examples

$f (x) = \sqrt[3]{x^{2}} + x \sqrt{x}$

Step 1

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 2

Set up the integral to solve.

$F (x) = \int \sqrt[3]{x^{2}} + x \sqrt{x} d x$

Step 3

Simplify.

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Step 3.1

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

$\int \sqrt[3]{x^{2}} + x \cdot x^{\frac{1}{2}} d x$

Step 3.2

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

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Step 3.2.1

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

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Step 3.2.1.1

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

$\int \sqrt[3]{x^{2}} + x^{1} x^{\frac{1}{2}} d x$

Step 3.2.1.2

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

$\int \sqrt[3]{x^{2}} + x^{1 + \frac{1}{2}} d x$

Step 3.2.2

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

$\int \sqrt[3]{x^{2}} + x^{\frac{2}{2} + \frac{1}{2}} d x$

Step 3.2.3

Combine the numerators over the common denominator.

$\int \sqrt[3]{x^{2}} + x^{\frac{2 + 1}{2}} d x$

Step 3.2.4

Add $2$ and $1$ .

$\int \sqrt[3]{x^{2}} + x^{\frac{3}{2}} d x$

Step 4

Split the single integral into multiple integrals.

$\int \sqrt[3]{x^{2}} d x + \int x^{\frac{3}{2}} d x$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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

Step 9

The answer is the antiderivative of the function $f (x) = \sqrt[3]{x^{2}} + x \sqrt{x}$ .

$F (x) =$ $\frac{3}{5} x^{\frac{5}{3}} + \frac{2}{5} x^{\frac{5}{2}} + C$