Need some help with this integral question could you include a video

Question

Use the Table of Integrals to evaluate the integral.

Answer 1

constants are factored out and ignored....

here is an outline of the itegration

1st step:

integrating arcsin(sqrt(x))

------------------------------

U = sqrt(x)

then U^2 = x

2 U du = dx

so the integral becomes:

arcsin(U) * U du

2nd step:

Next step is to integrate this new integral

in terms of U by parts:

Let M = arcsin(U) and dN = U

then dM = 1/sqrt(1 - U^2) and N = (1/2)U^2

the integral becomes:

(1/2)U^2 * arcsin U - (1/2) integral [ U^2 / sqrt( 1 - U^2) ]

3rd step:

next step is the change the var yet again...this time it's a

trig substitution:

Let U = sin T

then U^2 = (sinT)^2 and dU = cos T dT

note that the denominator becomes cos T which cancels the differential..

this integral becomes simply (sin T)^2

you can look this up in your table perhaps;

optionally, to finish it off, uses the identity

(sin T)^2 = (1/2)[1 - cos(2T)]

After all is said and done, the final answer is:

x arcsin(sqrt(x)) - (1/2) arcsin(sqrt(x)) + (1/4) sin ( 2 * arcsin(sqrt(x)) + C