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

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

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

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

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

Solution steps

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

Simplify

(\frac{x - 1}{x ^{2}})^{2} : \frac{( x - 1 ) ^{2}}{x ^{4}}

= \int \frac{( x - 1 ) ^{2}}{x ^{4}} d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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