Calculate integral (pie ---> pie^2) (cos `sqrt x` )/(`sqrt x` ) dx

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to evaluate the given definite integral, performing the following substitution, such that:

`sqrt x = t => 1/(2sqrt x)dx = dt => (dx)/(sqrt x) = 2dt`

Changing the limits of integration, yields:

`x = pi => t = sqrt pi`

`x = pi^2 => t = pi`

Replacing the variable yields:

`int_pi^(pi^2) (cos sqrt x)/(sqrt x) dx = int_(sqrt pi)^pi cos t*(2dt)`

`int_(sqrt pi)^pi cos t*(2dt) = 2 sin t|_(sqrt pi)^pi`

Using the fundamental theorem of calculus, yields:

`int_(sqrt pi)^pi cos t*(2dt) = 2(sin pi - sin sqrt pi)`

Since `sin pi = 0` yields:

`int_(sqrt pi)^pi cos t*(2dt) = -2sin sqrt pi`

Hence, evaluating the given definite integral, performing the indicated substitution, yields `int_pi^(pi^2) (cos sqrt x)/(sqrt x) dx = -2sin sqrt pi.`