How can I find $\int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$?

Question

My question is ; How can I solve the following integral question?

$\displaystyle \int_{\sqrt2/2}^{1}\int_{\sqrt{1-x^2}}^{x}\frac{1}{\sqrt{x^2+y^2}}dydx$

Thanks in advance,

Answer 1

You've had some time to study this, so let's look closer at the two evident approaches: (a) conversion to polar coordinates, (b) integrate directly with a standard hyperbolic substitution.

(a) Conversion to polar coordinates: Let

$$I = \int_{1/\sqrt{2}}^1 \, \int_{\sqrt{1-x^2}}^x \frac{1}{\sqrt{x^2+y^2}} \text{d}y \, \text{d}x.$$

If you draw a diagram of the situation you will see that

$$I = \int_0^{\pi/4} \, \int_{r=1}^{r=\sec \theta} \frac{1}{r} r \text{d}r \, \text{d} \theta = \int_0^{\pi/4} \sec \theta \, - 1 \textrm{ d} \theta $$

$$= \left[ \frac{1}{2} \log \left| \frac{1+ \sin \theta}{1- \sin \theta} \right| - \theta \right]_0^{\pi/4} = \log(1+\sqrt{2}) - \frac{\pi}{4}.$$

(b) Standard hyperbolic substitution: To evaluate the inner integral set $y= x \sinh u$ and noting that $\sinh^{-1} u = \log(u+ \sqrt{1+u^2})$ we have

$$\int_{\sqrt{1-x^2}}^x \frac{1}{\sqrt{x^2+y^2}} \text{d}y = \left[ \log \left( \frac{y}{x} + \sqrt{ 1+ \frac{y^2}{x^2} }\right) \right]_{\sqrt{1-x^2}}^x $$ $$= \log(1+ \sqrt{2}) + \log x - \log( 1+ \sqrt{1-x^2}). \quad (1)$$

Both the logs can be integrated by parts. The first is standard, $\int \log x \textrm{ d}x = x \log x - x + C$ and the second

$$ \int \log( 1+ \sqrt{1-x^2} ) \textrm{ d}x = x \log( 1+ \sqrt{1-x^2}) -x + \sin^{-1} x + C.$$

You will need (do the integration) to note that

$$\frac{1}{\sqrt{1-x^2}} - 1 = \frac{x^2}{(1-x^2) + \sqrt{1-x^2}}.$$

And so upon integrating $(1)$ between $1/\sqrt{2}$ and $1$ you obtain

$$I = \left(1 - \frac{1}{\sqrt{2}} \right) \log(1+ \sqrt{2}) + \left[ x \log x - x\log( 1+ \sqrt{1-x^2}) - \sin^{-1} x \right]_{1/\sqrt{2}}^1$$ $$= \log(1+\sqrt{2}) - \frac{\pi}{4}.$$

Answer 2

Try converting to polar coordinates, you should end out integrating $\sec(\theta) - 1$.

Answer 3

Hint:for the y integral, consider x to be a constant. It is a standard form that can be solved by a trigonometric substitution.

Answer 4

Have you tried matlab? If you say:

int('1/sqrt(x^2+y^2)','y','sqrt(1-x^2)','x')

then you'll get: log(x + (2*x^2)^(1/2)) - log((1 - x^2)^(1/2) + 1) as first integral.

Then: int('log(x + (2*x^2)^(1/2)) - log((1 - x^2)^(1/2) + 1)', 'x')

Result:

x*(log(x + (2*x^2)^(1/2)) - 1) - int(log((1 - x^2)^(1/2) + 1), x)

But the last part can't be integrated symbolically, so, in matlab, you can evaluate it numerically:

`double(int('log((1 - x^2)^(1/2) + 1)', 'x',sqrt(2)/2,1))` 

which will give : ans = 0.1143, and the other part you can evaluate yourself. It should be easy to replace x with the 2 numbers. Hope this helps!