For $a,b\in\mathbb{R}$ and $a, I'd like to evaluate the following improper integral $ \int_a^b \frac{dt}{\sqrt{(b-t)(t-a)}} $ Since I'd like to evaluate the integral rather than just make sure it converges, the comparison test isn't particularly useful. I started by expanding the terms under the radical, and then I attempted to complete the square. This left me with the following $ \int_a^b \frac{dt}{\sqrt{\left(t-\frac{a+b}{2}\right)^2 - \left(\frac{a+b}{2}\right)^2+ab}} $ In order for completing the square to be useful, I'd need to be able to make $ab-\left(\frac{a+b}{2}\right)^2$ into a square. I'll just sweep everything under the rug and say $ ab-\left(\frac{a+b}{2}\right)^2 = c^2 $ for some $c\in\mathbb{R}$. Similarly, I'll let $ u^2=\left(t-\frac{a+b}{2}\right)^2 $ Then my resulting integral has the form $ \int_a^b \frac{dt}{\sqrt{u^2-c^2}} $ I think at this point, I'm supposed to use a trig substitution to finish evaluating the integral. However, because of the substitutions I made above, I'm not really sure how to go about doing the actual substitution. Is there a better way to go about evaluating this integral, or do I just need to continue grinding it out?
Evaluating an improper integral with radical in denominator
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0@MichaelHardy You're right. Fixed. Thanks. – 2012-10-17
2 Answers
\begin{align} u & = \frac{t-a}{b-a} \\[8pt] dt & = (b-a)du \\[8pt] \sqrt{(b-t)(t-a)} & = (b-a)\sqrt{u(1-u)} \end{align} As $t$ goes from $a$ to $b$, then $u$ goes from $0$ to $1$. So $ \int_a^b\frac{dt}{\sqrt{(b-t)(t-a}} = \int_0^1 \frac{du}{\sqrt{u(1-u)}} = \int_0^1 u^{1/2-1} (1-u)^{1/2-1}\,du = B\left(\frac12,\frac12\right). $
$ =\frac{\Gamma\left(\frac12\right)\Gamma\left(\frac12\right)}{\Gamma\left(\frac12+\frac12\right)} = \pi. $
Later addendum: A standard identity says that if $ B(\alpha,\beta)=\int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\,dx $ and $ \Gamma(\alpha)=\int_0^\infty x^{\alpha-1} e^{-x}\,dx $ then $ B(\alpha,\beta)=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}. $ A proof is as follows. $ \Gamma(\alpha)\Gamma(\beta) = \int_0^\infty u^{\alpha-1}e^{-u}\,du \int_0^\infty v^{\beta-1}e^{-v}\,dv =\int_0^\infty\int_0^\infty u^{\alpha-1}v^{\beta-1}e^{-(u+v)}\,du\,dv.\tag{1} $ Now let $x=\frac{u}{u+v}, \qquad y=u+v,$ $dx\,dy=\left|\frac{\partial(x,y)}{\partial(u,v)}\right|\,du\,dv = \frac{du\,dv}{u+v} = \frac{du\,dv}{y}$ $ u = xy,\qquad v = (1-x)y. $ So $(1)$ becomes $ \int_0^\infty\int_0^1 (xy)^{\alpha-1} ((1-x)y)^{\beta-1} e^{-y} y\,dx\,dy $ $ = \int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\,dx\cdot\int_0^\infty y^{\alpha+\beta-1} e^{-y}\,dy = B(\alpha,\beta)\Gamma(\alpha+\beta). $
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0I've posted the question about $\Gamma(1/2)$. I posted an answer, and three other answers have already been posted. – 2012-10-17
Alternative approach:
$\frac{1}{\sqrt{(a-x)(b-x)}}=\left(\frac{1}{\sqrt{a-x}}+\frac{1}{\sqrt{b-x}}\right)\left(\frac{1}{\sqrt{a-x}+\sqrt{b-x}}\right)=-2\left(\frac{-1}{2\sqrt{a-x}}+\frac{-1}{2\sqrt{b-x}}\right)\left(\frac{1}{\sqrt{a-x}+\sqrt{b-x}}\right)$
Thus:
$\int_a^b \frac{1}{\sqrt{(a-x)(b-x)}}\;dx=-2\int_a^b\left(\frac{-1}{2\sqrt{a-x}}+\frac{-1}{2\sqrt{b-x}}\right)\left(\frac{1}{\sqrt{a-x}+\sqrt{b-x}}\right)\;dx$
$=-2\ln(\sqrt{a-x}+\sqrt{b-x})|_a^b=\ln\left(\frac{b-a}{a-b}\right)=i\pi$
Edit: noticed there has been a sign change under the square root in OP. This is equivalent to multiplying the above by $1/i$, so the result is now $\pi$, as Michael Hardy has shown.
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0Great. Thanks for all of your help. It's been a while since I've done a lot of integration like this and I'm having problems getting the intuition back, so it helps to hear how solutions were found. – 2012-10-17