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Calculus/limit of an integral function

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Expert: - 5/13/2010


Question
hi scotto... I just need some help regarding this question..
also please mention some basic theorems regarding dealing with such kind of questions.

lim      x^2   cos(t)^2 dt
         (     -----------
x-->0      )       xsinx
         0  
(
) here represents the sign of integration.
thanks

Answer
It is better to put it on one line, since it looks like

lim      x^2   cos(t)^2 dt
        (     -----------
x-->0      )       xsinx
        0  

Is this lim(x->0)((x²*cos²tdt)/(x*sinx))?

If that's the problem, cancel an x, giving lim(x->0)((x*cos²tdt)/(sinx)).
Next, note that lim(x->0)(sinx/x) = 1.
This occurs as x/sinx, and that is still 1.
So all we have left if cos²t dt.

Now if this is suppose to be integrated, note that cos²x = (1 + cos2x)/2.
Then note that ∫cos2x dx = sin(2x)/2 + C, so put that in the above formula.

If the problem is different, I await your response.

Calculus

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Scotto

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