1what is AOPS??
1) \int{\frac{{\sin (x+\alpha )}}{{\cos^{3}x}}}\sqrt{\frac{{\cos ecx+\sec x}}{{\cos ecx-\sec x}}}dx
2) \int{\frac{{x^{2}-1}}{{1+x^{2}}}}\frac{{dx}}{{\sqrt{1+x^{2}+x^{4}}}}
These questions have been lifted from AOPS .. I found it decent ...
There are a lot more which I will post shamelessly here :P
* I din know that AOPS was such a goldmine of good problems and discussions! (Now I know why it has so many fans!
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12 Answers
31) expanding sin(x+α) and dividing by cosx and then split the integrand into two ,then putting tanx=t the two integrands seems to be integrable.
3q2) on seeing the integral i think we can make a substitution
x+1/x=t
so dt=1-1/x2
and the terms inside the root will become t2-12
so final integral will be \int \frac{dt}{t(t^{2}-1^{2})^{1/2}}
11) \sqrt{(Cosecx+secx)/(cosecx-secx)} = \sqrt{(Cosx+ sinx)/(cosx-sinx)}
now, substitute \sqrt{(Cosx+ sinx)/(cosx-sinx)} = t
cn find a value of x frm here and then dx/dt. this might help
2) (x^2-1)/(1+x^2) dx/ \sqrt{1+x^2+x^4}
divide by x2 in both numerator and denominator.
=(1-1/x^2)/(1/x+x) dx/ \sqrt{1/x^2+1+x^2}
(1-1/x^2)/(1/x+x) dx/ \sqrt{(x+1/x)^2-1}
now, substitute ( x+ 1/x) = t. now, its integrable.