How do you evaluate the integral #int 1/sqrt(x-1)dx# from 5 to #oo#?

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Answer 1 · 2016-07-16T07:47:50

#int_(5)^(∞) (1)/(sqrt(x-1)) dx# diverges.

For this particular integral, we can do a u-substitution.

For #int_(5)^(∞) (1)/(sqrt(x-1)) dx#, let #u = x-1 -> du = dx#

Rewriting the integral gives us

#int_(5)^(∞) (1)/(u^(1/2)) du #

Actually, this integral does not converge by the p-series test, which tells us that

#sum_(n=1)^(∞) 1/n^(p)# converges if #p >1# and diverges if #p ≤ 1#.

In our case, since #p = 1/2#, which is surely #≤ 1#, we can conclude that this integral diverges.