anonymous
  • anonymous
Use trig substitution to solve the integral of 1/(4x^2(1-x^2)^(1/2)
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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hartnn
  • hartnn
so what substitution are you thinking ? any thoughts ?
anonymous
  • anonymous
I'm pretty sure you use \[x=asin (\Theta)\]
anonymous
  • anonymous
and a=1

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hartnn
  • hartnn
correct. tried it ?
anonymous
  • anonymous
so x=sin(theta) and dx=cos(theta) d(theta)
hartnn
  • hartnn
yup, what about 1-x^2 = .. ?
hartnn
  • hartnn
and (1-x^2)^1/2 =... ?
anonymous
  • anonymous
Looking at my work I got that that is cos(theta)
hartnn
  • hartnn
correct, so what does your integral turn into now ? everything in terms of theta
anonymous
  • anonymous
So then I plugged everything in and got \[\frac{ 1 }{ 4 }\int\limits_{}^{}\frac{ 1 }{ \sin ^{2}(\Theta)-\cos(\Theta) } d(\Theta)\]
hartnn
  • hartnn
4x^2 (1-x^2) would be 4 (sin theta)^2 cos theta right ? where did u get the - *minus* sign there ?
hartnn
  • hartnn
also your numerator is incorrect..
anonymous
  • anonymous
I don't know why I typed that minus sign.
anonymous
  • anonymous
I didn't get it. I just have sin^2cos
hartnn
  • hartnn
it just shouldn't be there. ... about the numerator, the dx becomes dx = cos theta d theta you missed cos theta term...
anonymous
  • anonymous
I sure did
anonymous
  • anonymous
so there should be a cos(theta) as the numerator
hartnn
  • hartnn
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
  • anonymous
Yep, got it
hartnn
  • hartnn
can u continue ?
anonymous
  • anonymous
couldn't you do (1/sin^2(theta) = csc^2(theta)?
hartnn
  • hartnn
absolutely, you can :)
anonymous
  • anonymous
then use the identity csc^2(theta) = -cot(theta)
hartnn
  • hartnn
don't forget the +c and then to resubstitute theta = sin^-1 x
anonymous
  • anonymous
theta = sin^-1x?
hartnn
  • hartnn
yes! x = sin theta so, theta = sin^-1 x inverse sin function
anonymous
  • anonymous
Ah, okay
anonymous
  • anonymous
\[\frac{ 1 }{ 4 }(-\cot(\sin^{-1} (x))?\]
hartnn
  • hartnn
yeah, i get the same, just with +c in the end
anonymous
  • anonymous
haha, sorry :P
anonymous
  • anonymous
I remember you have to draw a triangle at one point and solve for the missing side?
hartnn
  • hartnn
oh, if you need an algebraic answer instead of trigo one so, draw a right triangle first...
anonymous
  • anonymous
oh, okay. I think I remember now. Then sin(theta)=x/1
anonymous
  • anonymous
so then you just plug in the values and solve
hartnn
  • hartnn
right! correct way...if you get stuck, ask me
anonymous
  • anonymous
okay, awesome. Thank you!
hartnn
  • hartnn
welcome ^_^

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