Mathematics

Evaluate the definite integral :
$$\displaystyle \int_{1}^{2}\dfrac {1}{\sqrt {(x-1) (2-x)}}dx$$


SOLUTION
$$I=\displaystyle \int_{1}^{2} \dfrac {1}{\sqrt {-x^2 +3x-2}}dx =\displaystyle \int_{1}^{2} \dfrac {1}{\sqrt {-\left\{ \left(x-\dfrac {3}{2}\right)^2-\left(\dfrac {1}{2}\right)^2 \right\}}}dx =\displaystyle \int_{1}^{2} \dfrac {1}{\sqrt {\left (\dfrac {1}{2}\right)^2-\left (x-\dfrac {3}{2}\right)^2}}dx$$


Using,   $$\displaystyle \int \dfrac 1{\sqrt {a^2-x^2}}=\sin ^{-1}\dfrac xa $$

Replacing, we get

$$\displaystyle \int _1^2\dfrac 1{\sqrt {\left(\dfrac 12\right)^2-\left(x-\dfrac 32\right)^2}}$$

$$=\left| \sin ^{-1}\left(\dfrac {x-\dfrac {3}{2}}{\dfrac 12}\right)\right|_1^2$$

$$\left| \sin^{-1} (2x-3)\right|_1^2=\sin ^{-1}(1)-\sin ^{-1}(-1)$$

$$=\dfrac {\pi}2+\dfrac {\pi}2$$

$$=\pi$$
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