Calculus Examples

$f (x) = \frac{6}{{(4 - 3 x)}^{2}}$

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.

$F (x) = \int \frac{6}{{(4 - 3 x)}^{2}} d x$

Step 3

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \frac{1}{{(4 - 3 x)}^{2}} d x$

Step 4

Let $u = 4 - 3 x$ . 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 4.1

Let $u = 4 - 3 x$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $4 - 3 x$ .

$\frac{d}{d x} [4 - 3 x]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $4 - 3 x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 3 x]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- 3 x]$

Step 4.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

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

Step 4.1.3

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

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Step 4.1.3.1

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 4.1.3.2

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 4.1.3.3

Multiply $- 3$ by $1$ .

$0 - 3$

Step 4.1.4

Subtract $3$ from $0$ .

$- 3$

Step 4.2

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

$6 \int \frac{1}{u^{2}} \cdot \frac{1}{- 3} d u$

Step 5

Simplify.

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Step 5.1

Move the negative in front of the fraction.

$6 \int \frac{1}{u^{2}} (- \frac{1}{3}) d u$

Step 5.2

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{3}$ .

$6 \int - \frac{1}{u^{2} \cdot 3} d u$

Step 5.3

Move $3$ to the left of $u^{2}$ .

$6 \int - \frac{1}{3 u^{2}} d u$

Step 6

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

$6 (- \int \frac{1}{3 u^{2}} d u)$

Step 7

Multiply $- 1$ by $6$ .

$- 6 \int \frac{1}{3 u^{2}} d u$

Step 8

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

$- 6 (\frac{1}{3} \int \frac{1}{u^{2}} d u)$

Step 9

Simplify the expression.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{3}$ and $- 6$ .

$\frac{- 6}{3} \int \frac{1}{u^{2}} d u$

Step 9.1.2

Cancel the common factor of $- 6$ and $3$ .

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Step 9.1.2.1

Factor $3$ out of $- 6$ .

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

Step 9.1.2.2

Cancel the common factors.

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Step 9.1.2.2.1

Factor $3$ out of $3$ .

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

Step 9.1.2.2.2

Cancel the common factor.

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

Step 9.1.2.2.3

Rewrite the expression.

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

Step 9.1.2.2.4

Divide $- 2$ by $1$ .

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

Step 9.2

Apply basic rules of exponents.

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Step 9.2.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 9.2.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 9.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 2 \int u^{2 \cdot - 1} d u$

Step 9.2.2.2

Multiply $2$ by $- 1$ .

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

Step 10

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

$- 2 (- u^{- 1} + C)$

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Simplify.

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Step 11.2.1

Multiply $- 1$ by $- 2$ .

$2 \frac{1}{u} + C$

Step 11.2.2

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

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

Step 12

Replace all occurrences of $u$ with $4 - 3 x$ .

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

Step 13

The answer is the antiderivative of the function $f (x) = \frac{6}{{(4 - 3 x)}^{2}}$ .

$F (x) =$ $\frac{2}{4 - 3 x} + C$