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Cotx^3 积分

∫(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...

∫[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.

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

∫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

≡(1/2)Ln^2(sinx)+C

原式=∫csc^5xdx =∫csc^3x csc^2xdx =-∫csc^3xdcotx =-csc^3xcotx+∫cotx×3csc^2x×(-cscxcotx)dx =-csc^3xcotx-3∫(cot^2x×csc^...

see根特x,kosee根特x,ko探斤特x。 拓展: sinx: cosx = √(1 - sinx^2) tanx = sinx / √(1 - sinx^2) cotx = √(1 / sinx^2 - 1) secx = 1 / √(1 - sinx^2) cscx = 1 / sinx cosx: sinx = √(1 - cosx^2) tanx = √(1 / cosx^2 - 1) cotx = cosx / √(...

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