Calculus Examples

$f (x) = \sqrt[4]{x^{3}} + \sqrt[3]{x^{2}}$

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.

$F (x) = \int \sqrt[4]{x^{3}} + \sqrt[3]{x^{2}} d x$

Step 3

Split the single integral into multiple integrals.

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

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{3}}$ as $x^{\frac{3}{4}}$ .

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

Step 5

By the Power Rule, the integral of $x^{\frac{3}{4}}$ with respect to $x$ is $\frac{4}{7} x^{\frac{7}{4}}$ .

$\frac{4}{7} x^{\frac{7}{4}} + C + \int \sqrt[3]{x^{2}} d x$

Step 6

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

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

Step 7

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

$\frac{4}{7} x^{\frac{7}{4}} + C + \frac{3}{5} x^{\frac{5}{3}} + C$

Step 8

Simplify.

$\frac{4}{7} x^{\frac{7}{4}} + \frac{3}{5} x^{\frac{5}{3}} + C$

Step 9

The answer is the antiderivative of the function $f (x) = \sqrt[4]{x^{3}} + \sqrt[3]{x^{2}}$ .

$F (x) =$ $\frac{4}{7} x^{\frac{7}{4}} + \frac{3}{5} x^{\frac{5}{3}} + C$