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

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

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

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

x - 3 ln ∣ x ∣ - \frac{3}{x} + \frac{1}{2 x ^{2}} + C

Solution steps

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

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Expand

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

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

\int 1 d x = x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

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

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

= x - 3 ln ∣ x ∣ - \frac{3}{x} - (- \frac{1}{2 x ^{2}})

Simplify

= x - 3 ln ∣ x ∣ - \frac{3}{x} + \frac{1}{2 x ^{2}}

Add a constant to the solution

= x - 3 ln ∣ x ∣ - \frac{3}{x} + \frac{1}{2 x ^{2}} + C

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