Calculus Examples

$2 x (1 - x)$

Step 1

Write $2 x (1 - x)$ as a function.

$f (x) = 2 x (1 - x)$

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.

$F (x) = \int 2 x (1 - x) d x$

Step 4

Multiply $2 x (1 - x)$ .

$\int 2 x \cdot 1 + 2 x (- x) d x$

Step 5

Simplify.

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Step 5.1

Multiply $2$ by $1$ .

$\int 2 x + 2 x (- x) d x$

Step 5.2

Multiply $- 1$ by $2$ .

$\int 2 x - 2 x \cdot x d x$

Step 5.3

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

$\int 2 x - 2 (x^{1} x) d x$

Step 5.4

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

$\int 2 x - 2 (x^{1} x^{1}) d x$

Step 5.5

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

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

Step 5.6

Add $1$ and $1$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Simplify.

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Step 11.2.1

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

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

Step 11.2.2

Cancel the common factor of $2$ .

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Step 11.2.2.1

Cancel the common factor.

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

Step 11.2.2.2

Rewrite the expression.

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

Step 11.2.3

Multiply $x^{2}$ by $1$ .

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

Step 11.2.4

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

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

Step 11.2.5

Move the negative in front of the fraction.

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

Step 12

The answer is the antiderivative of the function $f (x) = 2 x (1 - x)$ .

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