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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: - \frac{( x - 1 ) ^{2}}{x ^{3}} - \int - \frac{2 ( x - 1 )}{x ^{4}} d x

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

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

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

Simplify

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

Add a constant to the solution

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

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