# Integrals : 2

• Nov 24th 2009, 06:00 AM
dhiab
Integrals : 2
Calculate integral for the function :

$\displaystyle \begin{array}{l} \ln (\sin (x)) \\ \ln (\cos (x)) \\ \end{array}$
• Nov 24th 2009, 07:21 AM
Jameson
Quote:

Originally Posted by dhiab
Calculate integral for the function :

$\displaystyle \begin{array}{l} \ln (\sin (x)) \\ \ln (\cos (x)) \\ \end{array}$

Mathematica got a nasty result for $\displaystyle \int \ln(\sin(x))dx$ involving i and non-elementary functions, so I don't know if I am misunderstanding what you are asked to do. If you meant a fraction, Mathematica can't solve it.
• Nov 24th 2009, 07:26 AM
Drexel28
Quote:

Originally Posted by dhiab
Calculate integral for the function :

$\displaystyle \begin{array}{l} \ln (\sin (x)) \\ \ln (\cos (x)) \\ \end{array}$

Quote:

Originally Posted by Jameson
Mathematica got a nasty result for $\displaystyle \int \ln(\sin(x))dx$ involving i and non-elementary functions, so I don't know if I am misunderstanding what you are asked to do. If you meant a fraction, Mathematica can't solve it.

Did the OP perhaps mean $\displaystyle \int\frac{\ln(\sin(x))}{\ln(\cos(x))}dx$?
• Nov 24th 2009, 07:32 AM
Jameson
Quote:

Originally Posted by Drexel28
Did the OP perhaps mean $\displaystyle \int\frac{\ln(\sin(x))}{\ln(\cos(x))}dx$?

I plugged that in as well and Mathematica couldn't find a solution. I haven't tried to work it out myself.
• Nov 24th 2009, 07:40 AM
Drexel28
Quote:

Originally Posted by Jameson
I plugged that in as well and Mathematica couldn't find a solution. I haven't tried to work it out myself.

Maybe we can make that a tad more apparent? Let $\displaystyle x=\arcsin(z)\implies dx=\frac{dz}{\sqrt{1-z^2}}$ so our integral becomes $\displaystyle \int\frac{\ln(z)}{\sqrt{1-z^2}\ln\left(\sqrt{1-z^2}\right)}$. Let $\displaystyle \sqrt{1-z^2}=\tau\implies \frac{-\tau}{\sqrt{1-\tau^2}}d\tau=dz$. So then our integral becomes $\displaystyle \frac{-1}{2}\int\frac{\ln\left(1-\tau^2\right)}{\sqrt{1-\tau^2}\ln\left(\tau\right)}d\tau$. That looks almost doable. I am not feeling it right now. Maybe the OP can continue.