Calc Help Pt. II: Integration, Limits & Averages

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Nimmy
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I need help in this problems. please :eek:

1. Integration of cos(lnx)

2. Integration of (csc^4 x/2) (cot x)

3. Integration of 2/Sqt. (4-x^2)

4. lim (e^x^2)-1/(2x^2)
x-)0

5. Find the average value of x/(x+3) [-a,a]
 
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Well we'd like to see what your thoughts are on these problems. I can offer some hints.

1) Use substitution
2) I'm not exactly sure what the question is
3) Use substitution
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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