1.∫(cotx)^2dx 2.设f(x)的一个原函数是xe^-x,则∫xf'(x)dx 3.∫(1-tanx)/(1+

1.∫(cotx)^2dx
=∫(cosx)^2/(sinx)^2dx
=∫[1-(sinx)^2]/(sinx)^2dx
=∫[(cscx)^2-1]dx
=-cotx-x+c
2.f(x)=x*e^(-x)=(1-x)e^(-x)
∫xf'(x)dx
=xf(x)-∫f(x)dx
=-x^2*[e^(-x)]+c
3.∫(1-tanx)/(1+tanx)dx
=∫tan(π/4-x)dx
=-∫sin(π/4-x)/cos(π/4-x)/d(π/4-x)
=lncos(π/4-x)+c (ln内取绝对值)
4.∫x*2^xdx
=x*(2^x)/ln2-∫(2^x)/ln2dx
=x*(2^x)/ln2-(2^x)/(ln2)^2+c
=(2^x)*(xln2-1)/(ln2)^2+c