Calculus Examples

$4 x - x^{2}$

Step 1

Write $4 x - x^{2}$ as a function.

$f (x) = 4 x - x^{2}$

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.

$F (x) = \int 4 x - x^{2} d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

Combine $4$ and $\frac{1}{2}$ .

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

Step 9.2.2

Cancel the common factor of $4$ and $2$ .

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Step 9.2.2.1

Factor $2$ out of $4$ .

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

Step 9.2.2.2

Cancel the common factors.

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Step 9.2.2.2.1

Factor $2$ out of $2$ .

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

Step 9.2.2.2.2

Cancel the common factor.

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

Step 9.2.2.2.3

Rewrite the expression.

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

Step 9.2.2.2.4

Divide $2$ by $1$ .

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

Step 10

The answer is the antiderivative of the function $f (x) = 4 x - x^{2}$ .

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