Calculus Examples

$f (x) = x (4 - x)$

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.

$F (x) = \int x (4 - x) d x$

Step 3

Multiply $x (4 - x)$ .

$\int x \cdot 4 + x (- x) d x$

Step 4

Simplify.

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Step 4.1

Move $4$ to the left of $x$ .

$\int 4 \cdot x + x (- x) d x$

Step 4.2

Raise $x$ to the power of $1$ .

$\int 4 x - (x^{1} x) d x$

Step 4.3

Raise $x$ to the power of $1$ .

$\int 4 x - (x^{1} x^{1}) d x$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 4 x - x^{1 + 1} d x$

Step 4.5

Add $1$ and $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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 7

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 8

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 9

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 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

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

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

Step 10.2.2

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

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Step 10.2.2.1

Factor $2$ out of $4$ .

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

Step 10.2.2.2

Cancel the common factors.

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Step 10.2.2.2.1

Factor $2$ out of $2$ .

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

Step 10.2.2.2.2

Cancel the common factor.

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

Step 10.2.2.2.3

Rewrite the expression.

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

Step 10.2.2.2.4

Divide $2$ by $1$ .

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

Step 11

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

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