Calculus Examples

$f (x) = {(3 x + 2)}^{4} - \frac{1}{x^{6}}$

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Let $u = 3 x + 2$ . 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 $u = 3 x + 2$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $3 x + 2$ .

$\frac{d}{d x} [3 x + 2]$

Step 4.1.2

By the Sum Rule, the derivative of $3 x + 2$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [2]$

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

$3 \frac{d}{d x} [x] + \frac{d}{d x} [2]$

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 $n = 1$ .

$3 \cdot 1 + \frac{d}{d x} [2]$

Step 4.1.3.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [2]$

Step 4.1.4

Differentiate using the Constant Rule.

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Step 4.1.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$3 + 0$

Step 4.1.4.2

Add $3$ and $0$ .

$3$

Step 4.2

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

$\int u^{4} \frac{1}{3} d u + \int - \frac{1}{x^{6}} d x$

Step 5

Combine $u^{4}$ and $\frac{1}{3}$ .

$\int \frac{u^{4}}{3} d u + \int - \frac{1}{x^{6}} d x$

Step 6

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

$\frac{1}{3} \int u^{4} d u + \int - \frac{1}{x^{6}} d x$

Step 7

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{3} (\frac{1}{5} u^{5} + C) + \int - \frac{1}{x^{6}} d x$

Step 8

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

$\frac{1}{3} (\frac{1}{5} u^{5} + C) - \int \frac{1}{x^{6}} d x$

Step 9

Apply basic rules of exponents.

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Step 9.1

Move $x^{6}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{3} (\frac{1}{5} u^{5} + C) - \int {(x^{6})}^{- 1} d x$

Step 9.2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 9.2.1

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

$\frac{1}{3} (\frac{1}{5} u^{5} + C) - \int x^{6 \cdot - 1} d x$

Step 9.2.2

Multiply $6$ by $- 1$ .

$\frac{1}{3} (\frac{1}{5} u^{5} + C) - \int x^{- 6} d x$

Step 10

By the Power Rule, the integral of $x^{- 6}$ with respect to $x$ is $- \frac{1}{5} x^{- 5}$ .

$\frac{1}{3} (\frac{1}{5} u^{5} + C) - (- \frac{1}{5} x^{- 5} + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

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Step 11.1.1

Combine $x^{- 5}$ and $\frac{1}{5}$ .

$\frac{1}{3} (\frac{1}{5} u^{5} + C) - (- \frac{x^{- 5}}{5} + C)$

Step 11.1.2

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{3} (\frac{1}{5} u^{5} + C) - (- \frac{1}{5 x^{5}} + C)$

Step 11.2

Simplify.

$\frac{1}{3} \cdot \frac{1}{5} u^{5} - - \frac{1}{5 x^{5}} + C$

Step 11.3

Simplify.

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Step 11.3.1

Multiply $\frac{1}{3}$ by $\frac{1}{5}$ .

$\frac{1}{3 \cdot 5} u^{5} - - \frac{1}{5 x^{5}} + C$

Step 11.3.2

Multiply $3$ by $5$ .

$\frac{1}{15} u^{5} - - \frac{1}{5 x^{5}} + C$

Step 11.3.3

Multiply $- 1$ by $- 1$ .

$\frac{1}{15} u^{5} + 1 \frac{1}{5 x^{5}} + C$

Step 11.3.4

Multiply $\frac{1}{5 x^{5}}$ by $1$ .

$\frac{1}{15} u^{5} + \frac{1}{5 x^{5}} + C$

Step 12

Replace all occurrences of $u$ with $3 x + 2$ .

$\frac{1}{15} {(3 x + 2)}^{5} + \frac{1}{5 x^{5}} + C$

Step 13

The answer is the antiderivative of the function $f (x) = {(3 x + 2)}^{4} - \frac{1}{x^{6}}$ .

$F (x) =$ $\frac{1}{15} {(3 x + 2)}^{5} + \frac{1}{5 x^{5}} + C$