How do you find the definite integral for: #(cos(sqrt(x)))/(sqrt(x))# for the intervals #[1, 4]#?

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Answer 1 · 2016-06-06T15:02:00

#2(sin(2)-sin(1))approx0.13565#

We have the integral:

#int_1^4cos(sqrtx)/sqrtxdx#

Use substitution. Let #u=sqrtx# and #du=1/(2sqrtx)dx#.

Multiply the integrand by #1/2# and the exterior of the integral by #2#.

#=2int_1^4cos(sqrtx)/(2sqrtx)dx=2int_1^4cos(sqrtx)(1/(2sqrtx))dx#

Now, make the substitutions. Recall that the bounds will change. The bound of #1# stays as #1# since #sqrt1=1#. The bound of #4# becomes #sqrt4=2#.

#=2int_1^2cos(u)du#

Note that #intcos(u)du=sin(u)+C#, so evaluate the integral:

#=2[sin(u)]_1^2=2[sin(2)-sin(1)]approx0.13565#