What is the integral of ((1-2x)/x)^3 ?

The integral of ((1-2x)/x)^3 is -1/(2x^2)+6/x+12ln|x|-8x+C

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integral of ((1-2x)/x)^3

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

- \frac{1}{2 x ^{2}} + \frac{6}{x} + 12 ln ∣ x ∣ - 8 x + C

Solution steps

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

Expand

(\frac{1 - 2 x}{x})^{3} : \frac{1}{x ^{3}} - \frac{6}{x ^{2}} + \frac{12}{x} - 8

= \int \frac{1}{x ^{3}} - \frac{6}{x ^{2}} + \frac{12}{x} - 8 d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{x ^{3}} d x - \int \frac{6}{x ^{2}} d x + \int \frac{12}{x} d x - \int 8 d x

\int \frac{1}{x ^{3}} d x = - \frac{1}{2 x ^{2}}

\int \frac{6}{x ^{2}} d x = - \frac{6}{x}

\int \frac{12}{x} d x = 12 ln ∣ x ∣

\int 8 d x = 8 x

= - \frac{1}{2 x ^{2}} - (- \frac{6}{x}) + 12 ln ∣ x ∣ - 8 x

Simplify

= - \frac{1}{2 x ^{2}} + \frac{6}{x} + 12 ln ∣ x ∣ - 8 x

Add a constant to the solution

= - \frac{1}{2 x ^{2}} + \frac{6}{x} + 12 ln ∣ x ∣ - 8 x + C

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What is the integral of ((1-2x)/x)^3 ?
The integral of ((1-2x)/x)^3 is -1/(2x^2)+6/x+12ln|x|-8x+C