Question

Evaluate $\underset{4}{\overset{9}{\int }}\frac{dx}{\sqrt{\left(9-x\right)\left(x-4\right)}}$

Answer 1

Consider the given integral.

$I={\int }_4^9\frac{dx}{\sqrt{(9-x)(x-4)}}$

$I={\int }_4^9\frac{dx}{\sqrt{9x-36-x^2+4x}}$

$I={\int }_4^9\frac{dx}{\sqrt{13x-36-x^2}}$

$I={\int }_4^9\frac{dx}{\sqrt{13x+\frac{169}{4}-\frac{169}{4}-36-x^2}}$

$I={\int }_4^9\frac{dx}{\sqrt{\frac{25}{4}-(-13x+\frac{169}{4}+x^2)}}$

$I={\int }_4^9\frac{dx}{\sqrt{{\left(\frac{5}{2}\right)}^2-{(\frac{13}{2}-x)}^2}}$

Let

$t=\frac{13}{2}-x$

$dt=-dx$

Therefore,

$I=-{\int }_{\frac{5}{2}}^{-\frac{5}{2}}\frac{dt}{\sqrt{{\left(\frac{5}{2}\right)}^2-t^2}}$

We know that

$\int \frac{dx}{\sqrt{a^2-x^2}}={\sin }^{-1}\left(\frac{x}{a}\right)+C$

Therefore,

$I=-{\left[{\sin }^{-1}\left(\frac{t}{\frac{5}{2}}\right)\right]}_{\frac{5}{2}}^{-\frac{5}{2}}$

$I=-[{\sin }^{-1}\left(\frac{2(-\frac{5}{2})}{5}\right)-{\sin }^{-1}\left(\frac{2\left(\frac{5}{2}\right)}{5}\right)]$

$I={\sin }^{-1}\left(1\right)-{\sin }^{-1}\left(1\right)$

$I=0$

Hence, the value is $0$.