Question

Improper Integral (Integrals with Infinite Limits)

∫dy/√(y-1) from 5 to ∞

Answer 1

Solution:

$$\int \frac{1}{\sqrt{y-1}}\,dx \quad from \quad 5\ to \ \infty$$

can be written as

$$\lim_{a \to \infty} \int_{5}^a \frac{1}{\sqrt{y-1}}\,dy$$

$$y-1 =t \implies\,dy=dt\\ y\rightarrow5; t\rightarrow4\ and\ y\rightarrow\infty; t\rightarrow\infty\\ \therefore \quad \lim_{a \to \infty} \int_{4}^a \frac{1}{\sqrt{t}}\,dt\\$$ $$\lim_{a \to \infty} \lbrack \lbrack \frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\rbrack _{a}-\lbrack \frac{t^{-\frac{1}{2}+1}}{-\frac{1}{2}+1}\rbrack _{4}\rbrack$$

$$\lim_{a \to \infty} \lbrack 2\sqrt{a} -2\sqrt{4}\rbrack \\ \lim_{a \to \infty} \lbrack 2\sqrt{a} -4\rbrack\\$$

As a tends to infinite, entire expression tends to infinite.

Thus the given integral is divergent.