Double integral question: Evaluating Integrals with Sinusoidal Functions

engineer_dave
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Double integral question...

Homework Statement



Evaluate the integrals shown ( I have attached the file with the integral).

Homework Equations





The Attempt at a Solution



Ok, for the first one, can you tell me how I integrate sin x^2...?? which method should i use? And for the second one, I don't really understand what the R is. What do u have to do to find the values next to to the integrand. Thanks!
 

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Can you integrate x*sin(x^2)?
 
yea but this doesn't have an x in front of it
 
any helpers lol??
 
Then I have no idea.
 
x is in the limits of the dy integration. Integrating sin(x^2) dy gives y*sin(x^2). Now put in the y limits THEN integrate dx. See, it does have an x in front of it.
 
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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