Algebra Examples

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

Step 1

The function $F (x)$ can be found by evaluating the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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

Step 8

The function $F$ if derived from the integral of the derivative of the function. This is valid by the fundamental theorem of calculus.

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