Calculus Examples

$f (x) = {(1 - 3 x)}^{2}$

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.

$F (x) = \int {(1 - 3 x)}^{2} d x$

Step 3

Let $u = 1 - 3 x$ . Then $d u = - 3 d x$ , so $- \frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 1 - 3 x$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $1 - 3 x$ .

$\frac{d}{d x} [1 - 3 x]$

Step 3.1.2

Differentiate.

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Step 3.1.2.1

By the Sum Rule, the derivative of $1 - 3 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x]$

Step 3.1.2.2

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

$0 + \frac{d}{d x} [- 3 x]$

Step 3.1.3

Evaluate $\frac{d}{d x} [- 3 x]$ .

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Step 3.1.3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$0 - 3 \frac{d}{d x} [x]$

Step 3.1.3.2

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

$0 - 3 \cdot 1$

Step 3.1.3.3

Multiply $- 3$ by $1$ .

$0 - 3$

Step 3.1.4

Subtract $3$ from $0$ .

$- 3$

Step 3.2

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

$\int u^{2} \frac{1}{- 3} d u$

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

$\int u^{2} (- \frac{1}{3}) d u$

Step 4.2

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

$\int - \frac{u^{2}}{3} d u$

Step 5

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

$- \int \frac{u^{2}}{3} d u$

Step 6

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$- (\frac{1}{3} \int u^{2} d u)$

Step 7

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

Simplify.

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Step 8.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

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

Step 8.2.2

Multiply $3$ by $3$ .

$- \frac{1}{9} u^{3} + C$

Step 9

Replace all occurrences of $u$ with $1 - 3 x$ .

$- \frac{1}{9} {(1 - 3 x)}^{3} + C$

Step 10

The answer is the antiderivative of the function $f (x) = {(1 - 3 x)}^{2}$ .

$F (x) =$ $- \frac{1}{9} {(1 - 3 x)}^{3} + C$