(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the integral of x * (arcsin x) * (1-x^{2})^{-1/2}dx

2. Relevant equations

integration by parts

3. The attempt at a solution

u = x, u' = 1

v' = (arcsin x) * (1-x^{2})^{-1/2}(= f(x) * f'(x) )

so v = ((arcsin x)^{2}) / 2

using integration by parts

uv - integral of u'v

= x * ((arcsin x)^{2}) / 2 - integral of ((arcsin x)^{2}) / 2

Now use integration by parts for a second time to find the new integral, taking the half out as a constant:

w = arcsin x w' = (1-x^{2})^{-1/2}

z' = arcsin x

so z = x * (arcsin x) + (1-x^{2})^{1/2}

wz - integral of w'z

= x * (arcsin x)^{2}+ (arcsin x) * (1-x^{2})^{1/2}- integral of [

x * (arcsin x) * (1-x^{2})-^{1/2}+ 1]

Substitute back into first equation (ie multiply above by -1/2)

integral of x * (arcsin x) * (1-x^{2})^{-1/2}dx =

x * ((arcsin x)^{2}) / 2 - x * ((arcsin x)^{2}) / 2 - ((arcsin x) * (1-x^{2})^{1/2}) / 2 + 1/2 * [integral of x * (arcsin x) * (1-x^{2})^{-1/2}dx] + 1/2 * integral of 1 dx

let [integral of x * (arcsin x) * (1-x^{2})^{-1/2}dx ] = I

I = - ((arcsin x) * (1-x^{2})^{1/2}) / 2 + I/2 + x/2

I/2 = - ((arcsin x) * (1-x^{2})^{1/2}) / 2 + x/2

I = - ((arcsin x) * (1-x^{2})^{1/2}) + x

However when I check this by differentiation I end up with - x * (arcsin x) * (1-x^{2})^{-1/2}. Hence I think the correct answer is ((arcsin x) * (1-x^{2})^{1/2}) - x

Thanks

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# Homework Help: Spot the mistake! integration

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