求不定积分(x^e^arctgx)/(1+x^2)^3/2dx提示：令arctgx=t

Question

求不定积分(x^e^arctgx)/(1+x^2)^3/2 dx 提示：令arctgx=t

Answer 1

=∫(x^e^arctgx)/(1+x^2)^1/2 d arctgx
=∫[(tg t)^e^t]/[1+(tg t)^2]^1/2 dt
=∫[(tg t)^e^t]·cost dt
=∫[(tg t)^e^t]·dsint
=sint·(tg t)^e^t-∫sint d[(tg t)^e^t]
=sint·(tg t)^e^t-∫(tg t)^e^t·sint·e^t·[ctg t +ln(tg t)] dt
=分部积分法