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Cotx^3 积分

解: ∫(cotx)³dx = ∫[cscx)² -1]*cotx*dx = ∫cotx*(cscx)²dx - ∫cosx/sinx *dx = - ∫cotx*d(cotx) - ∫1/sinx *d(sinx) = - 1/2 *cot²x - ln(|sinx|) +c

∫(cotx)^3dx =∫cotx*(cot^2x)dx =∫cotx*(csc^2x-1)dx =∫cotxdcotx-∫cotxdx 会算了吧?

I=∫(cotx)^3dx=∫(cosx)^3/(sinx)^3dx =∫(cosx)^2/(sinx)^3d(sinx) u=sinx, I=∫(1-u^2)/u^3 du=∫1/u^3 du- ∫1/udu =-1/2*u^(-2)-ln|u|+C

∫sinxdx/(sinx^3+cosx^3) =∫dx/sinx^2(1+cotx^3) =-∫dcotx/(1+cotx^3) cotx=u =-∫du/(1+u^3) =(-1/6)ln|u^2-u+1|+(1/√3)arctan[(2u-1)/√3] +(1/3)ln|u+1|+C =(-1/6)ln|cotx^2-cotx+1| +(1/√3)arctan[(2cotx-1)/√3]+(1/√3ln|cotx+1|+C ∫dx/(1+x^3...

∫(cscx )^3 dx =∫cscxd(-cotx)=-cscxcotx+∫cotxdcscx=-cscxcotx-∫cot²cscxdx=-cscxcotx-∫(csc³x-cscx)dx=-cscxcotx-∫csc³xdx+∫cscxdx=-cscxcotx-∫csc³xdx+ln|cscx-cotx|,所以∫(cscx )^3 dx =1/2(-cscxcotx+ln|csc...

∫[cotx/(sinx)^2]dx =∫[(cosx/sinx)/(sinx)^2]dx =∫[cosx/(sinx)^3]dx =∫[1/(sinx)^3]d(sinx) =-(1/2)[1/(sinx)^2]+C =-1/[2(sinx)^2]+C =-1/(1-cos2x)+C =1/(cos2x-1)+C.

两边同时对x求导: y'=3x^2cotx-3(csc^2x)x^3 所以dy=【3x^2cotx-3(csc^2x)x^3】dx

求不定积分∫[(xcosx)/(sin³x)]dx 解:原式=∫xcotxcsc²xdx=-∫xcotxd(cotx)=-(1/2)∫xd(cot²x)=-(1/2)[xcot²x-∫cot²xdx] =-(1/2)[xcot²x-∫(csc²x-1)dx]=-(1/2)[xcot²x+cotx+x]+C=-(1/2)[(1+cot²x)x+c...

∫1/(sinx)^4dx=∫(cscx)^2(cscx)^2dx =-∫(cscx)^2dcotx =-(cscx)^2cotx-∫cotx*2cscx*cotxcscxdx =-(cscx)^2cotx+2∫ (cotx)^2dcotx =-(cscx)^2cotx+(2/3)(cotx)^3+C

设t=3(tanx)^2 ∴(cotx)^2=3/t 原式=lim(t→0)(1+t)^(3/t) =[lim(t→0)(1+t)^(1/t)]^3 =e^3

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