How to integrate the following integral?∫cos√(x )dx

Int cos(sqrx) dx

First we
will assume that:

t = sqrt x ==> dt =
-1/2sqrx

                            =
-1/2tdx

               ==> dx =
-2tdt

==> Int cos(sqrtx) dx = Int cos(t) *-2tdt = -2
Int t*cos(t) dt...(1)

Now we will integrate t*cos(t)
t

Let  u = t ==> du =
dt

Let dv = cost dt ==> v =
sint

==> Int u dv = u*v - Int v
du

==> Int t*cost dt = t*sint - Int sint
dt

==> Int tcost dt = t*sint + cost
.........(2)

Now we will substitute (2) into
(1).

==> Int cos(sqrtx) dx = Int cos(t) *-2t dt = -2
Int t*cos(t) dt

==> Int cos(sqrtx)dx = -2[ t*sin(t)
+ cos(t)] + C

==> INt cos(sqrx) dx = -2t*sin(t)
-2cos(t) + c

Now we will substitute with t=
sqrtx

==> Int cos(sqrtx) dx =
-2sqrt(x)*sin(sqrtx) - 2cos(sqrtx) + C