Calculus Examples

$f (x) = \frac{4}{x^{5}} - {(1 - 2 x)}^{3}$

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.

$F (x) = \int \frac{4}{x^{5}} - {(1 - 2 x)}^{3} d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{4}{x^{5}} d x + \int - {(1 - 2 x)}^{3} d x$

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int \frac{1}{x^{5}} d x + \int - {(1 - 2 x)}^{3} d x$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

$4 \int {(x^{5})}^{- 1} d x + \int - {(1 - 2 x)}^{3} d x$

Step 5.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 5.2.1

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

$4 \int x^{5 \cdot - 1} d x + \int - {(1 - 2 x)}^{3} d x$

Step 5.2.2

Multiply $5$ by $- 1$ .

$4 \int x^{- 5} d x + \int - {(1 - 2 x)}^{3} d x$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Combine $x^{- 4}$ and $\frac{1}{4}$ .

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

Step 7.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 8

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

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

Step 9

Let $u = 1 - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 9.1

Let $u = 1 - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 9.1.1

Differentiate $1 - 2 x$ .

$\frac{d}{d x} [1 - 2 x]$

Step 9.1.2

Differentiate.

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Step 9.1.2.1

By the Sum Rule, the derivative of $1 - 2 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 2 x]$ .

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

Step 9.1.2.2

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

$0 + \frac{d}{d x} [- 2 x]$

Step 9.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 9.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$0 - 2 \frac{d}{d x} [x]$

Step 9.1.3.2

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

$0 - 2 \cdot 1$

Step 9.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 9.1.4

Subtract $2$ from $0$ .

$- 2$

Step 9.2

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

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

Step 10

Simplify.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

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

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Multiply $- 1$ by $- 1$ .

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

Step 12.2

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

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

Step 13

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 14

By the Power Rule, the integral of $u^{3}$ with respect to $u$ is $\frac{1}{4} u^{4}$ .

$4 (- \frac{1}{4 x^{4}} + C) + \frac{1}{2} (\frac{1}{4} u^{4} + C)$

Step 15

Simplify.

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Step 15.1

Simplify.

$\frac{- 1}{x^{4}} + \frac{1}{2} (\frac{1}{4} u^{4}) + C$

Step 15.2

Simplify.

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Step 15.2.1

Move the negative in front of the fraction.

$- \frac{1}{x^{4}} + \frac{1}{2} (\frac{1}{4} u^{4}) + C$

Step 15.2.2

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

$- \frac{1}{x^{4}} + \frac{1}{2} \cdot \frac{u^{4}}{4} + C$

Step 15.2.3

Multiply $\frac{1}{2}$ by $\frac{u^{4}}{4}$ .

$- \frac{1}{x^{4}} + \frac{u^{4}}{2 \cdot 4} + C$

Step 15.2.4

Multiply $2$ by $4$ .

$- \frac{1}{x^{4}} + \frac{u^{4}}{8} + C$

Step 16

Replace all occurrences of $u$ with $1 - 2 x$ .

$- \frac{1}{x^{4}} + \frac{{(1 - 2 x)}^{4}}{8} + C$

Step 17

Reorder terms.

$- \frac{1}{x^{4}} + \frac{1}{8} {(1 - 2 x)}^{4} + C$

Step 18

The answer is the antiderivative of the function $f (x) = \frac{4}{x^{5}} - {(1 - 2 x)}^{3}$ .

$F (x) =$ $- \frac{1}{x^{4}} + \frac{1}{8} {(1 - 2 x)}^{4} + C$