Calculus Examples

$f (x) = \frac{1}{\sqrt{42 - 3 x}}$

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.

$F (x) = \int \frac{1}{\sqrt{42 - 3 x}} d x$

Step 3

Let $u = 42 - 3 x$ . 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 $u = 42 - 3 x$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $42 - 3 x$ .

$\frac{d}{d x} [42 - 3 x]$

Step 3.1.2

Differentiate.

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Step 3.1.2.1

By the Sum Rule, the derivative of $42 - 3 x$ with respect to $x$ is $\frac{d}{d x} [42] + \frac{d}{d x} [- 3 x]$ .

$\frac{d}{d x} [42] + \frac{d}{d x} [- 3 x]$

Step 3.1.2.2

Since $42$ is constant with respect to $x$ , the derivative of $42$ 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 $d u$ .

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

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

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

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

Step 4.3

Move $3$ to the left of $\sqrt{u}$ .

$\int - \frac{1}{3 \sqrt{u}} d u$

Step 5

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

$- \int \frac{1}{3 \sqrt{u}} d u$

Step 6

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

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

Step 7

Apply basic rules of exponents.

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Step 7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Simplify.

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Step 9.2.1

Multiply $2$ by $- 1$ .

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

Step 9.2.2

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

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

Step 9.2.3

Move the negative in front of the fraction.

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

Step 10

Replace all occurrences of $u$ with $42 - 3 x$ .

$- \frac{2}{3} {(42 - 3 x)}^{\frac{1}{2}} + C$

Step 11

The answer is the antiderivative of the function $f (x) = \frac{1}{\sqrt{42 - 3 x}}$ .

$F (x) =$ $- \frac{2}{3} {(42 - 3 x)}^{\frac{1}{2}} + C$