Calculus Examples

$(4 - x) x^{- 3}$

Step 1

Write $(4 - x) x^{- 3}$ as a function.

$f (x) = (4 - x) x^{- 3}$

Step 2

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 3

Set up the integral to solve.

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

Step 4

Multiply $(4 - x) x^{- 3}$ .

$\int 4 x^{- 3} - x \cdot x^{- 3} d x$

Step 5

Multiply $x$ by $x^{- 3}$ by adding the exponents.

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Step 5.1

Move $x^{- 3}$ .

$\int 4 x^{- 3} - (x^{- 3} x) d x$

Step 5.2

Multiply $x^{- 3}$ by $x$ .

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Step 5.2.1

Raise $x$ to the power of $1$ .

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

Step 5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3

Add $- 3$ and $1$ .

$\int 4 x^{- 3} - x^{- 2} d x$

Step 6

Split the single integral into multiple integrals.

$\int 4 x^{- 3} d x + \int - x^{- 2} d x$

Step 7

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

$4 \int x^{- 3} d x + \int - x^{- 2} d x$

Step 8

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Simplify.

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

Step 12.2

Simplify.

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Step 12.2.1

Move the negative in front of the fraction.

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

Step 12.2.2

Multiply $- 1$ by $- 1$ .

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

Step 12.2.3

Multiply $x^{- 1}$ by $1$ .

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

Step 13

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

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