Question

求不定积分～谢谢了！！

Answer 1

设 x = tant，则 t = arctanx，dx = (sect)^2*dt
∫(x*arctanx)*dx/(1+x^2)^2
=∫tant*t*(sect)^2*dt/(sect)^4
=∫t*tant*dt/(sect)^2
=∫t*tant*(cost)^2*dt
=∫t*sint * cost *dt
=1/2*∫t*sin(2t)*dt
=1/2*[t*(-1/2*cos2t) + 1/2*∫cos2t*dt]
=-1/4*t*cos2t + 1/8*sin2t + C
=-1/4*arctanx *[2(cost)^2 -1] + 1/4 * sint *cost + C
=-1/4*arctanx *[2/(1+x^2) - 1] + 1/4 * tant * (cost)^2 + C
=-1/4*arctanx * (1-x^2)/(1+x^2) + 1/4*x * 1/(1+x^2) + C
=-1/4*arctanx * (1-x^2)/(1+x^2) + 1/4* x/(1+x^2) + C