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

1 Answer
Jul 16, 2016

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

Explanation:

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.