Calculus Examples

$f (x) = 5 e^{x} - 2 x^{4}$

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.

$F (x) = \int 5 e^{x} - 2 x^{4} d x$

Step 3

Split the single integral into multiple integrals.

$\int 5 e^{x} d x + \int - 2 x^{4} d x$

Step 4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int e^{x} d x + \int - 2 x^{4} d x$

Step 5

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

$5 (e^{x} + C) + \int - 2 x^{4} d x$

Step 6

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

$5 (e^{x} + C) - 2 \int x^{4} d x$

Step 7

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

$5 (e^{x} + C) - 2 (\frac{1}{5} x^{5} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$5 e^{x} - 2 (\frac{1}{5}) x^{5} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $- 2$ and $\frac{1}{5}$ .

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

Step 8.2.2

Move the negative in front of the fraction.

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

Step 9

The answer is the antiderivative of the function $f (x) = 5 e^{x} - 2 x^{4}$ .

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