Help in a primitive

• bnsm

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:

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

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!

Trig substitution is the obvious best choice here.

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:

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

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...

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

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...

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?