Question

求1除以根号下X(X+1)的不定积分

求大神。。

Answer 1

见图

Answer 2

I = integral 1/sqrt(x^2+x) dx
= integral 1/sqrt((x+1/2)^2-1/4) dx
substitute u = x+1/2 and du = dx:
I= integral 1/sqrt(u^2-1/4) du

substitute u = (sec(s))/2 and du = 1/2 tan(s) sec(s) ds.
Then sqrt(u^2-1/4) = sqrt((sec^2(s))/4-1/4) = (tan(s))/2 and s = sec^(-1)(2 u):

I= integral sec(s) ds
= ln(tan(s)+sec(s))+constant
Substitute back for s = sec^(-1)(2 u):
I = ln(sqrt(4 u^2-1)+2 u)+constant
Substitute back for u = x+1/2:
I = ln(2 x+2 sqrt(x (x+1))+1)+constant
= ln( x+2 sqrt(x (x+1))+(x+1) )+constant
=2 ln ( sqrt(x) + sqrt(x+1) ) + constant