Calculus Examples

$f (x) = - \frac{1}{{(6 x + 1)}^{2}}$

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.

$F (x) = \int - \frac{1}{{(6 x + 1)}^{2}} d x$

Step 3

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

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

Step 4

Let $u = 6 x + 1$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 6 x + 1$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $6 x + 1$ .

$\frac{d}{d x} [6 x + 1]$

Step 4.1.2

By the Sum Rule, the derivative of $6 x + 1$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [1]$

Step 4.1.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 4.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

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 $n = 1$ .

$6 \cdot 1 + \frac{d}{d x} [1]$

Step 4.1.3.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [1]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

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

$6 + 0$

Step 4.1.4.2

Add $6$ and $0$ .

$6$

Step 4.2

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

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

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1}{u^{2}}$ by $\frac{1}{6}$ .

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

Step 5.2

Move $6$ to the left of $u^{2}$ .

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

Step 6

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

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

Step 7

Apply basic rules of exponents.

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

Multiply $2$ by $- 1$ .

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

Step 8

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

$- \frac{1}{6} (- u^{- 1} + C)$

Step 9

Simplify.

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Step 9.1

Rewrite $- \frac{1}{6} (- u^{- 1} + C)$ as $- \frac{1}{6} (- \frac{1}{u}) + C$ .

$- \frac{1}{6} (- \frac{1}{u}) + C$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{6}) \frac{1}{u} + C$

Step 9.2.2

Multiply $\frac{1}{6}$ by $1$ .

$\frac{1}{6} \cdot \frac{1}{u} + C$

Step 9.2.3

Multiply $\frac{1}{6}$ by $\frac{1}{u}$ .

$\frac{1}{6 u} + C$

Step 10

Replace all occurrences of $u$ with $6 x + 1$ .

$\frac{1}{6 (6 x + 1)} + C$

Step 11

The answer is the antiderivative of the function $f (x) = - \frac{1}{{(6 x + 1)}^{2}}$ .

$F (x) =$ $\frac{1}{6 (6 x + 1)} + C$