Calculus Examples

$g' (x) = 3 x^{2} - 2 x - 4$

Step 1

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

$g (x) = \int g' (x) d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 8.1.2

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

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

Step 8.2

Simplify.

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

Step 9

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

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