Calculus Examples

$\frac{1}{\sqrt{x}} + \sqrt[3]{x}$

Step 1

Write $\frac{1}{\sqrt{x}} + \sqrt[3]{x}$ as a function.

$f (x) = \frac{1}{\sqrt{x}} + \sqrt[3]{x}$

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.

$F (x) = \int \frac{1}{\sqrt{x}} + \sqrt[3]{x} d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply basic rules of exponents.

Tap for more steps...

Step 5.1

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

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

Step 5.2

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 5.3

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

Tap for more steps...

Step 5.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 5.3.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$2 x^{\frac{1}{2}} + C + \int x^{\frac{1}{3}} d x$

Step 8

By the Power Rule, the integral of $x^{\frac{1}{3}}$ with respect to $x$ is $\frac{3}{4} x^{\frac{4}{3}}$ .

$2 x^{\frac{1}{2}} + C + \frac{3}{4} x^{\frac{4}{3}} + C$

Step 9

Simplify.

$2 x^{\frac{1}{2}} + \frac{3}{4} x^{\frac{4}{3}} + C$

Step 10

The answer is the antiderivative of the function $f (x) = \frac{1}{\sqrt{x}} + \sqrt[3]{x}$ .

$F (x) =$ $2 x^{\frac{1}{2}} + \frac{3}{4} x^{\frac{4}{3}} + C$