Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

$5 (\frac{1}{3} x^{3} + C) + \int - 5 x d x$

Step 6

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

$5 (\frac{1}{3} x^{3} + C) - 5 \int x d x$

Step 7

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

$5 (\frac{1}{3} x^{3} + C) - 5 (\frac{1}{2} x^{2} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$5 (\frac{1}{3}) x^{3} - 5 (\frac{1}{2}) x^{2} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $5$ and $\frac{1}{3}$ .

$\frac{5}{3} x^{3} - 5 (\frac{1}{2}) x^{2} + C$

Step 8.2.2

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

$\frac{5}{3} x^{3} + \frac{- 5}{2} x^{2} + C$

Step 8.2.3

Move the negative in front of the fraction.

$\frac{5}{3} x^{3} - \frac{5}{2} x^{2} + C$

Step 9

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

$F (x) =$ $\frac{5}{3} x^{3} - \frac{5}{2} x^{2} + C$