Mathematics

Solve $$\displaystyle \int{\dfrac{2x\ln \left( {{x}^{2}}-1 \right)}{\left( {{x}^{2}}-1 \right)}}dx$$


SOLUTION

Consider the given integral.


$$I=\displaystyle\int{\dfrac{2x\ln \left( {{x}^{2}}-1 \right)}{\left( {{x}^{2}}-1 \right)}}dx$$


 


Let $$t=\ln \left( {{x}^{2}}-1 \right)$$


$$ \dfrac{dt}{dx}=\dfrac{1}{{{x}^{2}}-1}\times 2x $$


$$ dt=\dfrac{2x}{{{x}^{2}}-1}dx $$


 


Therefore,


$$ I=\displaystyle\int{tdt} $$


$$ I=\dfrac{{{t}^{2}}}{2}+C $$


 


On putting the value of $$t$$, we get


$$I=\dfrac{{{\left( \ln \left( {{x}^{2}}-1 \right) \right)}^{2}}}{2}+C$$


 


Hence, this is the answer.

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Subjective Medium Published on 17th 09, 2020
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