求不定积分∫(sinx)/(si
∫(sinx)/(sin^3(x)+cos^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)...全部
∫(sinx)/(sin^3(x)+cos^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。
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