Calculus Examples

$f (x) = 7 x^{2} - x$

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.

$F (x) = \int 7 x^{2} - x d x$

Step 3

Split the single integral into multiple integrals.

$\int 7 x^{2} d x + \int - x d x$

Step 4

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int x^{2} d x + \int - x d x$

Step 5

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

$7 (\frac{1}{3} x^{3} + C) + \int - x d x$

Step 6

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

$7 (\frac{1}{3} x^{3} + C) - \int x d x$

Step 7

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

$7 (\frac{1}{3} x^{3} + C) - (\frac{1}{2} x^{2} + C)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Simplify.

$7 (\frac{1}{3}) x^{3} - \frac{1}{2} x^{2} + C$

Step 8.2

Combine $7$ and $\frac{1}{3}$ .

$\frac{7}{3} x^{3} - \frac{1}{2} x^{2} + C$

Step 9

The answer is the antiderivative of the function $f (x) = 7 x^{2} - x$ .

$F (x) =$ $\frac{7}{3} x^{3} - \frac{1}{2} x^{2} + C$