I have to evaluate the improper integral 1/sqrt(x)*e^(-sqrt(x)) from 0 to pi and I don't know how to find my answer. Any help will be greatly appreciated!``

Expert Answers
embizze eNotes educator| Certified Educator

We can rewrite the integrand using laws of exponents:

`=int_(0)^(pi) (e^(sqrt(x))dx)/(sqrt(x)) `

Note that the derivative of `e^(sqrt(x)) ` is `e^(sqrt(x)) d/(dx)[sqrt(x)] `

or `1/(2sqrt(x))e^(sqrt(x)) `

Multiply the integral by 2 and the integrand by 1/2 to get:

`=2 int_(0)^(pi) (e^(sqrt(x))dx)/(2sqrt(x)) `  So the integrand is in the form `e^udu ` with `u=sqrt(x),du=1/(2sqrt(x))dx ` and the integral of e^udu=e^u so

`=2e^(sqrt(x)) |_(0)^(pi) `

`=2e^(sqrt(pi))-2e^0 `

`=2e^(sqrt(pi))-2 ~~ 9.7705545 `


`int_(0)^(pi) 1/(sqrt(x)e^(-sqrt(x)))dx=2e^(sqrt(pi))-2~~9.7705545 `


Unless there is more to this problem, this is an ordinary definite integral. Improper integrals might have infinity as one or both of the limits of integration, or the interval of integration might contain a finite number of discontinuities, or there might be some problem like a discontinuity at an endpoint, etc...

You can use a u-substitution to evaluate the integral, but you must make sure to change the limits of integration.