# Thread: Integration Problem.

1. ## Integration Problem.

Find the integral of
{{(x4 + 1)1/2 }[ ln( x2 + 1 ) + 2lnx]} /x​4

2. ## Re: Integration Problem.

Originally Posted by sohamjariwala
Find the integral of
{{(x4 + 1)1/2 }[ ln( x2 + 1 ) + 2lnx]} /x​4
It doesn't have a closed form solution. Maybe try a series solution instead...

3. ## Re: Integration Problem.

I think I made a mistake in writing it

the correct question is
{{x2 + 1)1/2[ ln(x2 + 1) - 2lnx]}/x​4

4. ## Re: Integration Problem.

$\text{Integrate:}\mspace{7mu} \int{\frac{\sqrt{x^2+1}}{x^4}}\left(\ln{(x^2+1)}-2\ln{x}\right)dx}\quad \quad (1)$

$(1)\, \Leftrightarrow\, \int{\frac{\sqrt{x^2+1}}{x^4}}\ln\left({\frac{x^2+ 1}{x^2}}\right)dx}\, \Leftrightarrow\, \int{\frac{1}{x^3}\sqrt{\frac{x^2+1}{x^2}}\ln\left ({\frac{x^2+1}{x^2}}\right)dx}$

$\text{Let}\mspace{6mu} u=\sqrt{\frac{x^2+1}{x^2}}\, \Leftrightarrow\, u^2=\frac{x^2+1}{x^2}=1+\frac{1}{x^2}>0$

$\Rightarrow\, 2udu=-\frac{2}{x^3}dx\, \Leftrightarrow\, dx=-x^3udu$

$\int{\frac{1}{x^3}\cdot u\cdot \ln{u^2}\cdot (-x^3udu)}\, \Leftrightarrow\, -2\int{u^2\ln{(u)}du}\quad \quad (2)$

$\text{Integration by parts}$

$\begin{array}{lllllll} w=\ln{u} &&& & & dv=u^2du\\ dw=\dfrac{1}{u} &&& & & v=\dfrac{u^3}{3}\\ \end{array}$

$(2)\, \Leftrightarrow\, -\frac{2}{3}\left( u^3\ln{u}-\int{u^2}du \right) = -\frac{2}{9}u^3 \left( 3\ln{u}-1 \right) = -\frac{2}{9}u^3 \left( \ln{\frac{u^3}{e}} \right)$

$\text{No need to have the absolute value sign for \emph{u} in the log term since}\,u>0.$

$\Leftrightarrow\, -\frac{2}{9} \left( \frac{x^2+1}{x^2}\right)^{3/2} \ln{\left[ \frac{1}{e}\left(\frac{x^2+1}{x^2}\right)^{3/2} \right]}$

$\Leftrightarrow\, -\frac{2}{9} \cdot \frac{\left(x^2+1\right)^{3/2}}{x^3} \ln{\frac{\left(x^2+1\right)^{3/2}}{ex^3}}\, \Leftrightarrow\, \frac{2}{9} \cdot \frac{\left(x^2+1\right)^{3/2}}{x^3} \ln{\frac{ex^3}{\left(x^2+1\right)^{3/2}}}$

$\int{\frac{\sqrt{x^2+1}}{x^4}}\left(\ln{(x^2+1)}-2\ln{x}\right)dx} = \frac{2}{9} \cdot \frac{\left(x^2+1\right)^{3/2}}{x^3} \ln{\frac{ex^3}{\left(x^2+1\right)^{3/2}}} + C$

$\text{Time for some tea!}$

5. ## Re: Integration Problem.

Superb,My friend thank you.

6. ## Re: Integration Problem.

I also have another problem. Can you figure out what to substitute.
Integrate w.r.t. x
1/[(cot(x/2)cot(x/3)cot(x/6)]