Use trig substitution to solve the integral of 1/(4x^2(1-x^2)^(1/2)

Question

hartnn · Answer

so what substitution are you thinking ? any thoughts ?

anonymous · Answer

I'm pretty sure you use $$x=asin (\Theta)$$

anonymous · Answer

and a=1

hartnn · Answer

correct.
tried it ?

anonymous · Answer

so x=sin(theta) and dx=cos(theta) d(theta)

hartnn · Answer

yup, 
what about 1-x^2 = .. ?

hartnn · Answer

and (1-x^2)^1/2 =... ?

anonymous · Answer

Looking at my work I got that that is cos(theta)

hartnn · Answer

correct,
so what does your integral turn into now ? 
everything in terms of theta

anonymous · Answer

So then I plugged everything in and got $$\frac{ 1 }{ 4 }\int\limits_{}^{}\frac{ 1 }{ \sin ^{2}(\Theta)-\cos(\Theta) } d(\Theta)$$

hartnn · Answer

4x^2 (1-x^2) would be 
4 (sin theta)^2 cos theta
right ?

where did u get the - *minus* sign there ?

hartnn · Answer

also your numerator is incorrect..

anonymous · Answer

I don't know why I typed that minus sign.

anonymous · Answer

I didn't get it.  I just have sin^2cos

hartnn · Answer

it just shouldn't be there. ...
about the numerator, the dx becomes 
dx = cos theta d theta
you missed cos theta term...

anonymous · Answer

I sure did

anonymous · Answer

so there should be a cos(theta) as the numerator

hartnn · Answer

yup, which gets cancelled with the denominator cos theta
so you are left with just 
integral 1/ sin^2 theta dtheta 
ok til here ?

anonymous · Answer

Yep, got it

hartnn · Answer

can u continue ?

anonymous · Answer

couldn't you do (1/sin^2(theta) = csc^2(theta)?

hartnn · Answer

absolutely, you can :)

anonymous · Answer

then use the identity csc^2(theta) = -cot(theta)

hartnn · Answer

don't forget the +c 
and then to resubstitute theta = sin^-1 x

anonymous · Answer

theta = sin^-1x?

hartnn · Answer

yes!

x = sin theta
so, theta = sin^-1 x 
inverse sin function

anonymous · Answer

Ah, okay

anonymous · Answer

$$\frac{ 1 }{ 4 }(-\cot(\sin^{-1} (x))?$$

hartnn · Answer

yeah, i get the same, just with +c in the end

anonymous · Answer

haha, sorry :P

anonymous · Answer

I remember you have to draw a triangle at one point and solve for the missing side?

hartnn · Answer

oh, if you need an algebraic answer instead of trigo one
so, draw a right triangle first...

anonymous · Answer

oh, okay.  I think I remember now.  Then sin(theta)=x/1

anonymous · Answer

so then you just plug in the values and solve

hartnn · Answer

right!
correct way...if you get stuck, ask me

anonymous · Answer

okay, awesome.  Thank you!

hartnn · Answer

welcome ^_^