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

1 Answer
Jun 6, 2016

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

Explanation:

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