Question

Integrate :$\int {({\left(\frac{x+1}{x-1}\right)}^2+{\left(\frac{x-1}{x+1}\right)}^2-2)}^{\frac{1}{2}}dx$

Answer 1

b'

$\int {({\left(\frac{x+1}{x-1}\right)}^2+{\left(\frac{x-1}{x+1}\right)}^2-2)}^{\frac{1}{2}}dx$

$=int{left\left({left(dfracx+1x-1+dfracx-1x+1right)}^{2}right\right)}^{frac12}dx$

$displaystyleintleft(dfracx+1x-1+dfracx+1x+1right)dx$

$=displaystyleintdfrac{x}^2-1+x^2-1{left(x-1right)}^{2}dx$

$=displaystyleintdfrac2x^2{x}^2-1dx$

$=displaystyleintdfrac2x^2-2+2x^2-1dx$

$=displaystyleintdfrac2x^2-2x^2-1dx+displaystyleintdfrac2x^2-1dx$

$=2displaystyleintdfrac{x}^2-1x^2-1dx+displaystyleintdfrac2x^2-1dx$

$=2displaystyleintdx+displaystyleintdfrac2x^2-1dx$

$=2left\left[xright\right]+left\left[lnleft\left(dfracx-1x+1right\right)right\right]+c$ where $displaystyleintdfracdx{x}^2-a^2=dfrac12alnleft\left(dfracx-ax+aright\right)+c$

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