Calculus Examples

$\sqrt[5]{x} - 2 e^{x}$

Step 1

Write $\sqrt[5]{x} - 2 e^{x}$ as a function.

$f (x) = \sqrt[5]{x} - 2 e^{x}$

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.

$F (x) = \int \sqrt[5]{x} - 2 e^{x} d x$

Step 4

Split the single integral into multiple integrals.

$\int \sqrt[5]{x} d x + \int - 2 e^{x} d x$

Step 5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\int x^{\frac{1}{5}} d x + \int - 2 e^{x} d x$

Step 6

By the Power Rule, the integral of $x^{\frac{1}{5}}$ with respect to $x$ is $\frac{5}{6} x^{\frac{6}{5}}$ .

$\frac{5}{6} x^{\frac{6}{5}} + C + \int - 2 e^{x} d x$

Step 7

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$\frac{5}{6} x^{\frac{6}{5}} + C - 2 \int e^{x} d x$

Step 8

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$\frac{5}{6} x^{\frac{6}{5}} + C - 2 (e^{x} + C)$

Step 9

Simplify.

$\frac{5}{6} x^{\frac{6}{5}} - 2 e^{x} + C$

Step 10

The answer is the antiderivative of the function $f (x) = \sqrt[5]{x} - 2 e^{x}$ .

$F (x) =$ $\frac{5}{6} x^{\frac{6}{5}} - 2 e^{x} + C$