求下列不定积分： (1)∫[(x/√(x2+1)-x√(x2+1)]dx (2)∫dx/xlnx； (3)∫exdx/√(1-e

求下列不定积分：
(1)∫[(x/√(x²+1)-x√(x²+1)]dx
(2)∫dx/xlnx；
(3)∫e^xdx/√(1-e^2x)；
(4)∫1/1+x²e^arctanxdx．
【正确答案】：(1)∫[x/√(x²+1)-x√(x²+1)]dx =1/2∫[1/√(x²+1)]d(x²+1)-(1/2)∫√(x²+1)d(x²+1) =√(x²+1)-(1/2)×(2/3)(x³+1)^3/2+C =√(x²+1)-(1/3)(x²+1)^3/2+C (2)∫dx/xlnx=∫(1/lnx)d(lnx) =In|lnx|+C． (3)∫e^xdx/√1-e^2x =∫1/√1-e^2xde^x =arcsine^x+C． (4)∫1/1+x²e^arctanxdx =∫e^arctanxdarctanx =e^arctanx+C