Question

1/(tant-1)sec²t的积分

Answer 1

问：1/(tant-1)sec²t的积分

答：ctanx=t, 则 x=tant, dx=sec^2tdt ∫xe^arctanx/(1+x^2)^3/2dx =∫tante^t/(1+tan^2;t)^3/2*sec^2tdt =∫tante^t/sec ^2t sec ^2tdt =∫tante^t/sectdt =∫sinte^tdt (1) =-∫e^tdcost =-coste^t+∫coste^tdt =-coste^t+sinte^t-∫sinte^tdt (2) 由 (1)(2)得 ∫sinte^tdt =1/2( sinte^t-coste^t ) +C =1/2( sint-cost)e^t +C =1/2cost(tant-1)e^t +C =1/2*1/√(tan^2t+1)*(tant-1)e^t +C =1/2*1/√(x^2+1)*(x-1)e^arctanx+C =√(x^2+1)*(x-1)e^arctanx/(x^2+1)+C 即 ∫xe^arctanx/(1+x^2)^3/2dx =√(x^2+1)*(x-1)e^arctanx/(x^2+1)+C