Changing variable integral mmn15 1B

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\int_{0}^{5}\int_{0}^{\sqrt{{25-x^{2}}}}ln(1+x^{2}+y^{2})dxdy
how to solve it?
 
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You might want to try converting it into polar coordinates:

x=rcosθ
y=rsinθ

For the limits you would need to see what shape they are describing.
 
what about dx dy
 
What does your textbook say about integrating in polar coordinates?
 

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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