Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 6

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

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

Step 7

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

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

Step 8

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

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

Step 9

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

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

Step 10

Simplify.

Tap for more steps...

Step 10.1

Simplify.

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

Step 10.2

Simplify.

Tap for more steps...

Step 10.2.1

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

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

Step 10.2.2

Combine $2$ and $\frac{x^{2}}{2}$ .

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

Step 10.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.2.3.1

Cancel the common factor.

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

Step 10.2.3.2

Divide $x^{2}$ by $1$ .

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

Step 11

Reorder terms.

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

Step 12

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

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