Calculus Examples

$f (r) = \frac{3}{1 + r^{2}} - \sqrt[5]{r^{4}}$

Step 1

The function $F (r)$ can be found by finding the indefinite integral of the derivative $f (r)$ .

$F (r) = \int f (r) d r$

Step 2

Set up the integral to solve.

$F (r) = \int \frac{3}{1 + r^{2}} - \sqrt[5]{r^{4}} d r$

Step 3

Split the single integral into multiple integrals.

$\int \frac{3}{1 + r^{2}} d r + \int - \sqrt[5]{r^{4}} d r$

Step 4

Since $3$ is constant with respect to $r$ , move $3$ out of the integral.

$3 \int \frac{1}{1 + r^{2}} d r + \int - \sqrt[5]{r^{4}} d r$

Step 5

Rewrite $1$ as $1^{2}$ .

$3 \int \frac{1}{1^{2} + r^{2}} d r + \int - \sqrt[5]{r^{4}} d r$

Step 6

The integral of $\frac{1}{1^{2} + r^{2}}$ with respect to $r$ is $\arctan (r) + C$ .

$3 (\arctan (r) + C) + \int - \sqrt[5]{r^{4}} d r$

Step 7

Since $- 1$ is constant with respect to $r$ , move $- 1$ out of the integral.

$3 (\arctan (r) + C) - \int \sqrt[5]{r^{4}} d r$

Step 8

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{r^{4}}$ as $r^{\frac{4}{5}}$ .

$3 (\arctan (r) + C) - \int r^{\frac{4}{5}} d r$

Step 9

By the Power Rule, the integral of $r^{\frac{4}{5}}$ with respect to $r$ is $\frac{5}{9} r^{\frac{9}{5}}$ .

$3 (\arctan (r) + C) - (\frac{5}{9} r^{\frac{9}{5}} + C)$

Step 10

Simplify.

$3 \arctan (r) - \frac{5}{9} r^{\frac{9}{5}} + C$

Step 11

The answer is the antiderivative of the function $f (r) = \frac{3}{1 + r^{2}} - \sqrt[5]{r^{4}}$ .

$F (r) =$ $3 \arctan (r) - \frac{5}{9} r^{\frac{9}{5}} + C$