Calculus Examples

$\sqrt{x} + \sqrt[3]{x} + \sqrt[4]{x}$

Step 1

Write $\sqrt{x} + \sqrt[3]{x} + \sqrt[4]{x}$ as a function.

$f (x) = \sqrt{x} + \sqrt[3]{x} + \sqrt[4]{x}$

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.

$F (x) = \int \sqrt{x} + \sqrt[3]{x} + \sqrt[4]{x} d x$

Step 4

Split the single integral into multiple integrals.

$\int \sqrt{x} d x + \int \sqrt[3]{x} d x + \int \sqrt[4]{x} d x$

Step 5

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

$\int x^{\frac{1}{2}} d x + \int \sqrt[3]{x} d x + \int \sqrt[4]{x} d x$

Step 6

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

$\frac{2}{3} x^{\frac{3}{2}} + C + \int \sqrt[3]{x} d x + \int \sqrt[4]{x} d x$

Step 7

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

$\frac{2}{3} x^{\frac{3}{2}} + C + \int x^{\frac{1}{3}} d x + \int \sqrt[4]{x} d x$

Step 8

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

$\frac{2}{3} x^{\frac{3}{2}} + C + \frac{3}{4} x^{\frac{4}{3}} + C + \int \sqrt[4]{x} d x$

Step 9

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

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

Step 10

By the Power Rule, the integral of $x^{\frac{1}{4}}$ with respect to $x$ is $\frac{4}{5} x^{\frac{5}{4}}$ .

$\frac{2}{3} x^{\frac{3}{2}} + C + \frac{3}{4} x^{\frac{4}{3}} + C + \frac{4}{5} x^{\frac{5}{4}} + C$

Step 11

Simplify.

$\frac{2}{3} x^{\frac{3}{2}} + \frac{3}{4} x^{\frac{4}{3}} + \frac{4}{5} x^{\frac{5}{4}} + C$

Step 12

The answer is the antiderivative of the function $f (x) = \sqrt{x} + \sqrt[3]{x} + \sqrt[4]{x}$ .

$F (x) =$ $\frac{2}{3} x^{\frac{3}{2}} + \frac{3}{4} x^{\frac{4}{3}} + \frac{4}{5} x^{\frac{5}{4}} + C$