Help in a primitive

[bnsm]

Help in a primitive!

Homework Statement

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:

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

Homework Equations

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!

 

[quantumdude]

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:

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

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?