用第一类换元积分法求下列不定积分1、∫ln^3x/xdx2、∫1/x^2-9dx3、∫1/√1-4x^2dx

Question

用第一类换元积分法求下列不定积分
1、∫ln^3x/x dx
2、∫1/x^2-9 dx
3、∫1/√1-4x^2 dx

Answer 1

1.原式=∫ln^3xd(lnx)
=(ln^4x)/4+C
2.原式=∫1/[(x+3)(x-3)]dx
=1/6*∫(1/(x-3)-1/(x+3))dx
=1/6∫d(x-3)/(x-3)-1/6∫d(x+3)/(x+3)
=1/6ln|x-3|-1/6ln|x+3|+C
=1/6ln|(x-3)/(x+3)|+C
3.原式=1/2∫d(2x)/√(1-(2x)^2)
=1/2arcsin(2x)+C