∫dxsin3xcos5x=

I=∫dxsin3xcos5x Dividing Nr∈Dr by cos4⁡x=∫sec4⁡xsin3⁡xcos5⁡xcos8⁡xdx=∫sec4⁡xtan3⁡xdx=∫sec2⁡xsec2⁡xtan2⁡x⋅tan⁡xdx=∫1+tan2⁡xsec2⁡xtan⁡x⋅tan⁡xdx Put tan⁡x=tsec2⁡xdx=dt=∫1+t2dttt=∫t−32+t12dt=t−32+1−32+1+t12+112+1+c=t−12−12+t3232+c=+2−1t⋅+tt3+c=2−3+t23t+c=23t2−3t+c=23tan2⁡x−3tan⁡x+c