Question

44题怎么做，求解答

Answer 1

你好 令arctanx=t，则x=tant，dx=sec²tdt ∫xe^arctanx/(1+x^2)^3/2dx =∫tante^t/(1+tan^2;t)^3/2*sec²tdt =∫tante^t/sec ³t 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/√(tan²t+1)*(tant-1)e^t +C =1/2*1/√(x²+1)*(x-1)e^arctanx+C =√(x²+1)*(x-1)e^arctanx/(x²+1)+C 即 ∫xe^arctanx/(1+x^2)^3/2dx =√(x²+1)*(x-1)e^arctanx/(x²+1)+C