calculus 2 trig substitution ∫ √(1-x^2)/x^4 dx?

Question

I need to use trig substitution to compute the following integral:
 ∫ √(1-x^2)/x^4 dx
I know it equals -1/3(√(1-x^2)/x)^3 + C
so far; 
a=1
x=sin(θ)
dx=cos(θ) dθ
1-x^2=cos^2 θ
my teacher advised me to draw a triangle which I did, and after substituting my integral is:
=∫ (√(cos^2 (θ))/sin^4 θ) cos(θ)dθ
=∫ ((cos^2 (θ) dθ)/sin^4 θ)
=∫ ((cos^2 (θ))/sin^2 θ) (1/sin^2 (θ)) dθ
{after u-substitution}
=-∫ cot^2(θ) -csc^2(θ) dθ
Now I know I should use the power rule and then back substitue, but I'm not sure where the 1/3 comes from in the final answer, and also if I need to use any more trig identities. 
Thanks very much for any assistance, I'll choose the best answer. Thanks again

Anonymous · Accepted Answer

Note that the derivative of cotθ is -csc^2θ, so let:
u = cotθ ==> du = -csc^2θ dθ.

Then, the integral becomes:
- ∫ cot^2θ(-csc^2θ) dθ = - ∫ cot^2θ (-csc^2θ dθ)
= - ∫ u^2 du, by applying substitutions
= (-1/3)u^3 + C, by integrating
= (-1/3)cot^3θ + C, by back-substituting u = cotθ.
(Note that this is where the 1/3 comes into play.)

At this point, use x = sinθ and draw your triangle to get cotθ in terms of x. You should get:
cotθ = √(1 - x^2)/x.

Back-substituting gives the required answer.

I hope this helps!

? · Answer

Hmm... once you enable x= 4sin? then x^2=sixteen(sin?)^2 then if youin case you element out the sixteen from the oblong root, you have have been given: ? 4x³ ?(one million-(sin?)^2) you may grant the 4 out in front of the essential like this: 4? x³ ?(one million-(sin?)^2) there might desire to be the trig identity: (sin?)^2 + (cos?)^2=one million (jointly as you progression the sin?^2 to the different factor you get what's below the sq. root) next, which you would be able to say one million-sin?^2=cos?^2. the oblong root of it is cos?. Now, your necessary is 4? x³ cos? d? use integration via components: u=x³ du= 3x^2 and dv= cos? v=-sin? using the form uv - ?v du, keeep doing this different than you get to an crucial variety that seems familar to you. Then plug interior the better and slash limits of the mandatory you ought to get 2048/15 as your respond desire this helped :) Ohh i forgot to assert... it fairly is doable to do integration by skill of components greater beneficial than as quickly as and the variables don't have have been given to be a similar... purely the variety they are in. (as an occasion in case you chosen u=cos the common time you may no longer substitute it to be x^2 or something the 2d time... It besides the undeniable fact that must be in trig variety)

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calculus 2 trig substitution ∫ √(1-x^2)/x^4 dx?

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