Question

what is the integral od sin2x/csc^2 X sec^2 X ? ..ty :D

Answer 1

I'm assuming you're saying
\[\int\frac{\sin 2x}{\csc^2x\sec^2x}~dx\]
Use the double angle identity to rewrite the numerator:
\[\int\frac{2\sin x\cos x}{\csc^2x\sec^2x}~dx\]
Since sine/cosecant and cosine/secant are reciprocal function, you have
\[2\int\sin^3 x\cos^3 x~dx\]
Reduce the power of either sine or cosine. I'll do it with the cosine:
\[2\int\sin^3 x\cos^2 x\cos x~dx\]
Pythagorean identity:
\[2\int\sin^3 x(1-\sin^2x)\cos x~dx\]
Distribute
\[2\left(\int\sin^3 x\cos x~dx-\int\sin^2x\cos x~dx\right)\]
Solve each integral with the substitution \(u=\sin x\).