设x=sect
√(x²-1)=tant
d(sect)=sec(t)*tan(t)dt
∫[√(x²-1) ]dx
=∫tan(t)sec(t)*tan(t)dt
=∫sin^2(t)/cos^3(t)tdt
=∫[1-cos^2(t)]/cos^3(t)dt
=∫1/cos^3(t)dt - ∫1/costdt
=(x*(x^2 - 1)^(1/2))/2 - log(x + (x^2 - 1)^(1/2))/2
不定积分的公式:
1、∫ a dx = ax + C,a和C都是常数
2、∫ x^a dx = [x^(a + 1)]/(a + 1) + C,其中a为常数且 a ≠ -1
3、∫ 1/x dx = ln|x| + C
4、∫ a^x dx = (1/lna)a^x + C,其中a > 0 且 a ≠ 1
5、∫ e^x dx = e^x + C
6、∫ cosx dx = sinx + C
7、∫ sinx dx = - cosx + C
8、∫ cotx dx = ln|sinx| + C = - ln|cscx| + C