Question

求解∫1/(cos^4(x)sin^2(x))dx

Answer 1

∫1/[(cosx)^4(sinx)]dx=∫[(sinx)^2+(cosx)^2]/[(cosx)^4(sinx)]dx

=∫(secx)^4dx+4∫(csc2x)^2dx

∫(secx)^4dx=∫(secx)^2[(tanx)^2+1]dx=∫[(tanx)^2+1]dtanx=(tanx)^3/3+tanx

∫(csc2x)^2dx=-1/2*cot2x

所以∫1/[(cosx)^4(sinx)]dx=(tanx)^3/3+tanx-2cot2x+C