Calculus Examples

$\frac{x}{2} - 4 x^{3} - x^{4}$

Step 1

Write

$\frac{x}{2} - 4 x^{3} - x^{4}$ as a function.

$f (x) = \frac{x}{2} - 4 x^{3} - x^{4}$

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.

$F (x) = \int \frac{x}{2} - 4 x^{3} - x^{4} d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

Since

$\frac{1}{2}$ is constant with respect to

$x$ , move

$\frac{1}{2}$ out of the integral.

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

Step 6

By the Power Rule, the integral of

$x$ with respect to

$x$ is

$\frac{1}{2} x^{2}$ .

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

Step 7

Since

$- 4$ is constant with respect to

$x$ , move

$- 4$ out of the integral.

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

Step 8

By the Power Rule, the integral of

$x^{3}$ with respect to

$x$ is

$\frac{1}{4} x^{4}$ .

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 4 (\frac{1}{4} x^{4} + C) + \int - x^{4} d x$

Step 9

Since

$- 1$ is constant with respect to

$x$ , move

$- 1$ out of the integral.

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 4 (\frac{1}{4} x^{4} + C) - \int x^{4} d x$

Step 10

By the Power Rule, the integral of

$x^{4}$ with respect to

$x$ is

$\frac{1}{5} x^{5}$ .

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 4 (\frac{1}{4} x^{4} + C) - (\frac{1}{5} x^{5} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{x^{2}}{4} - x^{4} - (\frac{1}{5} x^{5}) + C$

Step 11.2

Combine

$\frac{1}{5}$ and

$x^{5}$ .

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

Step 12

Reorder terms.

$\frac{1}{4} x^{2} - x^{4} - \frac{1}{5} x^{5} + C$

Step 13

The answer is the antiderivative of the function

$f (x) = \frac{x}{2} - 4 x^{3} - x^{4}$ .

$F (x) =$

$\frac{1}{4} x^{2} - x^{4} - \frac{1}{5} x^{5} + C$