MHB Solving $\displaystyle \int \frac{\ln\left(x^2+2\right)}{(x+2)^2}dx$ by Parts

[juantheron]

$\displaystyle \int \frac{\ln\left(x^2+2\right)}{(x+2)^2}dx$

$\bf{My\; Try::}$ Given $\displaystyle \int \ln \left(x^2+2\right)\cdot \frac{1}{(x+2)^2}dx$

Using Integration by parts, we get

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

Is there is any other method by which we can solve the Integral

$\displaystyle \int \frac{x}{\left(x^2+2\right)\cdot (x+2)}dx$ other then partial fraction.

Help me

Thanks

 

[MarkFL]

Why the aversion to partial fraction decomposition? I think it is the best course of action for continuing. :D

 

[Prove It]

I agree with Mark, the best strategy would be partial fractions.

Try $\displaystyle \begin{align*} \frac{A\,x + B}{x^2 + 2} + \frac{C}{x + 2} \equiv \frac{x}{ \left( x^2 + 2 \right) \left( x + 2 \right) } \end{align*}$ for your partial fraction decomposition.

 

[juantheron]

Thanks Markfl and prove it.

 

[Prometheus1]

While partial fractions is fine/the standard way, asking for an alternative way is something I feel should always be encouraged (and never discouraged) in mathematics. Looking for alternative ways to do problems builds one's creativity far better than sticking to the usual methods.