Help in a primitive!!!

[bnsm]

1. The problem statement, all variables and given/known data

Hello guys! Please, I'm really needing help in a primitive... I don't know, maybe it has a simple solution, but I'm tired and blocked on this... Can you give some lights? Here goes the equation:

\int\frac{dx}{x^{2}\sqrt{4-x^{2}}}

2. Relevant equations



3. The attempt at a solution

I tried substitution of 4-x^2 and of x^2, but none of them work... I also tried by parts, with u'=1/(x^2) and v=1/sqrt(4-x^2), but it looks like it becomes even heavier... Can you help me?

Thanks to all and to this great site!

[Tom Mattson]

Trig substitution is the obvious best choice here.

[bnsm]

Trig substitution is the obvious best choice here.

Yes, of course, you're right! Many Thanks! :) I made x=2*sin(t) and I got:

\int\frac{dt}{4sin^{2}\left(t\right)}

[bnsm]

Ok, I'm stucked again... I tried:

\frac{1}{4}\int\frac{sin^{2}\left(t\right)+cos^{2} \left(t\right)}{sin^{2}\left(t\right)}dt

which gave:

\frac{t}{4}+\int\frac{cos^{2}\left(t\right)}{sin^{ 2}\left(t\right)}dt

Any ideas? I tried partial and substitution but it's a mess...

[Dick]

Try and differentiate cot(x)=cos(x)/sin(x), ok? What do you get?

[bnsm]

Try and differentiate cot(x)=cos(x)/sin(x), ok? What do you get?

I substituted the fraction above by the cot(t) and then I made the primitive by parts, considering

u'=1 and thus u=t
v=cot(t) and thus v'=-2cot(t)/((sin(t))^2)

Then, I tried to develop the following:

\int\frac{cos^{2}\left(t\right)}{sin^{2}\left(t\ri ght)}=t\cot^{2}\left(t\right)+\int\frac{2t\cot\lef t(t\right)}{sin^{2}\left(t\right)}

What do you think about this? I can try to substitute cot(t) by cos(t)/sin(t), but I'll get a (sin(x))^3 in the denominator... The point is that it seems I'm getting a primitive even more complicated...

[Dick]

You are making this way too complicated. You wanted to find the integral of dt/sin(t)^2. All I was trying to point out is that the derivative of cot(t) is -1/sin(t)^2. Doesn't that make it easy?