Find the most general antiderivative of #f(x)=((1+x)/x)^2#?

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#f(x)=((1+x)/x)^2#

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Answer 1 · 2017-11-20T21:27:19

#int f(x) dx = x + ln(x^2) - 1/x + C#

Given:

#int ((1+x)/x)^2dx#

Expand the square:

#int (x^2+2x+1)/x^2dx#

Perform the division:

#int 1+2/x+1/x^2dx#

Separate into 3 integrals:

#int dx +2int 1/x dx +int 1/x^2dx#

These integrals are well known:

#x + 2ln|x| - 1/x + C#

Bring the 2 inside the logarithm:

#x + ln(x^2) - 1/x + C#