Evaluating improper integral $ \int_a^b \frac{dt}{\sqrt{(b-t)(t-a)}} $

Question

For $a,b\in\mathbb{R}$ and $a<b$, 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?

Answer 1

Apply the Euler-like substitution $\sqrt{(x-a)(b-x)} =(x-\frac{b+a}2)t$ to get $$dt= \frac{-(b-a)^2\ dx}{(a+b-2x)^2\sqrt{(x-a)(b-x)}} ,\>\>\> 1+t^2=\frac{(b-a)^2}{(a+b-2x)^2}$$ Thus $$\int_a^b\frac1{\sqrt{(x-a)(b-x)} }dx=\int_{-\infty}^\infty \frac1{1+t^2}dt=\pi $$

Answer 2

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.

Answer 3

\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 $$ \int_0^\infty\int_0^1 (xy)^{\alpha-1} ((1-x)y)^{\beta-1} e^{-y} y\,dx\,dy $$ $$ = \int_0^1 x^{\alpha-1}(1-x)^{\beta-1}\,dx\cdot\int_0^\infty y^{\alpha+\beta-1} e^{-y}\,dy = B(\alpha,\beta)\Gamma(\alpha+\beta). $$

Answer 4

$$ \begin{align} \int_a^b\frac{\mathrm{d}t}{\sqrt{(b-t)(t-a)}} &=\int_0^{b-a}\frac{\mathrm{d}t}{\sqrt{(b-a-t)t}}\tag{1a}\\ &=\sqrt{b-a}\int_0^1\frac{\mathrm{d}t}{\sqrt{(1-t)t}}\tag{1b}\\ &=\sqrt{b-a}\,\frac{\Gamma\!\left(\frac12\right)\Gamma\!\left(\frac12\right)}{\Gamma(1)}\tag{1c}\\[12pt] &=\pi\sqrt{b-a}\tag{1d} \end{align} $$ Explanation:
$\text{(1a):}$ substitute $t\mapsto t+a$
$\text{(1b):}$ substitute $t\mapsto(b-a)t$
$\text{(1c):}$ Beta Integral (formula shown here)
$\text{(1d):}$ $\Gamma\!\left(\frac12\right)=\sqrt\pi$