分类：教育/科学-理工学科-数学
问题：求不定积分∫（sinx）/(sin^3(x)+cos^3(x)) 
∫（sinx）/(sin^3(x)+c^3(x))dx∫（cosx）/(sin^3(x)+cos^3(x))dx
回答：∫（sinx）/(sin^3(x)+c^3(x))dx=∫(sinx)/[sin^3(x)(1+cot^3(x))]dx=∫csc^2(x)/(1+cot^3(x)dx=-∫1/(1+cot^3(x)dcotx, (记u=cotx)=(-1/3){∫1/(u+1)du-∫(u-2)/(u^2-u+1)du}=(-1/3){ln|1+u|-1/2∫(2u-1)/(u^2-u+1)du+3/2∫1/(u^2-u+1)du}=(-1/3){ln|1+u|-1/2ln(u^2-u+1)+√3arctan(u-1/2)}+C.∫（cosx）/(sin^3(x)+cos^3(x))dx=∫(cosx)/[cos^3(x)(tan^3(x)+1)]dx=∫sec^2(x)/[tan^3(x)+1]dx=∫1/[(tan^3(x)+1]dtanx, (记v=tanx)=(1/3){∫1/(v+1)dv-∫(v-2)/(v^2-v+1)dv}=(1/3){ln|1+v|-1/2∫(2v-1)/(v^2-v+1)dv+3/2∫1/(v^2-v+1)dv}=(1/3){ln|1+v|-1/2ln(v^2-v+1)+√3arctan(v-1/2)}+C.