<s>[INST] Evaluating a Discontinuous Integral How do you evaluate this integral? $$\int _0 ^4\frac{dx}{x^2 - 2x - 3}$$
My work:
The expression $x^2 - 2x - 3$ is discontinuous at $x = 3$ in the interval $x = 0$ to $x = 4$, so I got to integrate the expression like this:
$$\int _0 ^4\frac{dx}{x^2 - 2x - 3}$$ is equal to
$$\int _0 ^3\frac{dx}{x^2 - 2x - 3} + \int _3 ^4\frac{dx}{x^2 - 2x - 3} $$.
Then recall that in evaluating a discontinuous integrand, the integral is defined by the relations $$\int_a ^b f(x) = lim_{x->b^-} \int_a ^x f(x) dx $$
if $x = b$  is the discontinuous point or $$\int_a ^b f(x) = lim_{x->a^+} \int_x ^b f(x) dx $$
if $x = a$  is the discontinuous point.
With that in mind:
$$\int _0 ^3\frac{dx}{x^2 - 2x - 3} + \int _3 ^4\frac{dx}{x^2 - 2x - 3} $$
$$\int _0 ^3\frac{dx}{(x^2 - 2x + 1) + (-3)(-1)} + \int _3 ^4\frac{dx}{(x^2 - 2x + 1) + (-3)(-1)} $$
$$\int _0 ^3\frac{dx}{(x-1)^2 -(2)^2} + \int _3 ^4\frac{dx}{(x-1)^2  -(2)^2} $$
Remembering that $\int \frac{du}{u^2 +a^2} = \frac{1}{a} \arctan \left( \frac{u}{a}\right) + C, $
we get:
$$\int _0 ^3\frac{dx}{(x-1)^2 -(2)^2} +\int _3 ^4\frac{dx}{(x-1)^2 -(2)^2}$$
equals
$$lim_{x->3^+} \int_0 ^x \frac{dx}{(x-1)^2 -(2)^2} + lim_{x->3^-} \int_x ^4 \frac{dx}{(x-1)^2 -(2)^2} $$
equals
$$lim_{x->3^+} \left( \frac{1}{2} \arctan \left( \frac{x-2}{2}\right) \right)|_3 ^x + lim_{x->3^-} \left(\frac{1}{2} \arctan \left( \frac{x-2}{2}\right) \right)|_x ^4$$
equals
$$\left ( \frac{1}{2} \arctan \left( \frac{x-2}{2}\right)-\frac{1}{2} \arctan \left( \frac{3-2}{2}\right) \right ) + \left ( \frac{1}{2} \arctan \left( \frac{4-2}{2}\right)-\frac{1}{2} \arctan \left( \frac{x-2}{2}\right) \right )$$
equals
$$\left ( \frac{1}{2} \arctan \left( \frac{x-2}{2}\right)- 0.2318 \right ) + \left ( \frac{\pi}{8}-\frac{1}{2} \arctan \left( \frac{x-2}{2}\right) \right )$$
which makes  $$\int _0 ^4\frac{dx}{x^2 - 2x - 3} = 0.1609$$
But in my book, it said there is no value of $\int _0 ^4\frac{dx}{x^2 - 2x - 3}.$
How do you prove that the integral $$\int _0 ^4\frac{dx}{x^2 - 2x - 3}$$ has no value?
 [/INST] $$I=\int _0 ^4\frac{dx}{x^2 - 2x - 3} =\frac{1}{4} \int _0 ^4\left(\frac{1}{x - 3}- \frac{1}{x+1}\right)dx$$
As already pointed out, there is a singularity at $x=3$.
So, at the common sens, the integral isn't convergent. But this is not true in the sens of Cauchy integration (Cauchy Principal Value : http://mathworld.wolfram.com/CauchyPrincipalValue.html )
$$I=\frac{1}{4}\lim_{\epsilon\to 0} \left( \int _0 ^{3-\epsilon}\frac{dx}{x - 3} +\int _{3+\epsilon}^4\frac{dx}{x - 3} - \int _0^4\frac{dx}{x+1}\right)$$
$\int _0 ^{3-\epsilon}\frac{dx}{x - 3} +\int _{3+\epsilon}^4\frac{dx}{x - 3} = \left[\ln|x-3| \right]_{x=0}^{x=3-\epsilon} +\left[\ln|x-3| \right]_{x=3+\epsilon}^{x=4} = \ln(\epsilon)-\ln(3)+\ln(1)-\ln(\epsilon) = -\ln(3)$
$\int _0^4\frac{dx}{x+1}=\left[\ln|x+1| \right]_{x=0}^4=\ln(4+1)-\ln(1)=\ln(5)$
$$\text{PV}\int _0 ^4\frac{dx}{x^2 - 2x - 3} = \frac{1}{4}\left(-\ln(3)-\ln(5)\right)=-\frac{\ln(15)}{4}$$
The symbol PV means that the integral is considered on the sens of a Cauchy integral. Indeed, the integral as a finite value, the so called Principal Value.
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