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

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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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

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