First slide

Evaluation of definite integrals

Question

_{$\int_{2}^{3} \frac{2 - x}{\sqrt{5 x - 6 - x^{2}}} d x =$}

Easy

A

_{$\frac{π}{2}$}
B

_{$- \frac{π}{2}$}
C

_{$- π$}
D

_$π$

Solution

$\int_{2}^{3} \frac{2 - x}{\sqrt{5 x - 6 - x^{2}}} d x$
$= \frac{1}{2} \int_{2}^{3} \frac{(5 - 2 x) - 1}{\sqrt{5 x - 6 - x^{2}}} d x$
$= \frac{1}{2} \int_{2}^{3} \frac{5 - 2 x}{\sqrt{5 x - 6 - x^{2}}} d x - \frac{1}{2} \int_{2}^{3} \frac{1}{\sqrt{{(\frac{1}{2})}^{2} - {(x - \frac{5}{2})}^{2}}} d x$
$= \frac{1}{2} {(2 \sqrt{5 x - 6 - x^{2}})}_{2}^{3} - \frac{1}{2} S i n^{- 1} {(\frac{x - \frac{5}{2}}{\frac{1}{2}})}_{2}^{3}$
$= - \frac{π}{2}$

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