Calculus Examples

$f (x) = 2 {(x - 1)}^{2}$

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.

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

Step 3

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

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

Step 4

Let $u = x - 1$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x - 1$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x - 1$ .

$\frac{d}{d x} [x - 1]$

Step 4.1.2

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [- 1]$

Step 4.1.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$1 + 0$

Step 4.1.5

Add $1$ and $0$ .

$1$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$2 \int u^{2} d u$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Rewrite $2 (\frac{1}{3} u^{3} + C)$ as $2 (\frac{1}{3}) u^{3} + C$ .

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

Step 6.2

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

$\frac{2}{3} u^{3} + C$

Step 7

Replace all occurrences of $u$ with $x - 1$ .

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

Step 8

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

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

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