求不定积分

∫（sinx）/(sin^3(x)+cos^3(x))dx∫（cosx）/(sin^3(x)+cos^3(x))dx

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2013-02-17

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    ∫（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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