Calculus Examples

$x^{2} - 3 x$

Step 1

Write $x^{2} - 3 x$ as a function.

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Simplify.

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

Step 8.2

Simplify.

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Step 8.2.1

Combine $- 3$ and $\frac{1}{2}$ .

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

Step 8.2.2

Move the negative in front of the fraction.

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

Step 9

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

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