How to integrate this problem?
I don't know where should I start with this one.
$\displaystyle \int\frac{dx}{\tan2x[1-2lnsin2x]}$
Just in case a picture helps...
As people have hinted...
... i.e., notice how to differentiate $\displaystyle 1 - 2\ln(\sin2x)$ wrt x. I.e. consider how to differentiate $\displaystyle 1 - 2\ln t$ wrt t, or the corresponding balloon here, wrt the dashed balloon...
But then apply the chain rule, where (key in spoiler) ...
Spoiler:
So we get...
And notice that for this we also used...
Now for the integration... we want to travel up instead of down. Can you anti-differentiate
$\displaystyle \frac{-1}{4t}$
wrt t? I.e...
If so then you're ready to fill out the rest of this...
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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