Evaluate: (x-3)(5-x) d x - Vidyadip

MCQ

Evaluate: $\int \sqrt{(x-3)(5-x)} d x$

A
$\frac{1}{2}(x-4) \sqrt{(x-3)(5-x)}+\frac{1}{2} \cos ^{-1}(x-4)+C$
✓
$\frac{1}{2}(x-4) \sqrt{(x-3)(5-x)}+\frac{1}{2} \sin ^{-1}(x-4)+C$
C
$\frac{1}{2} \sqrt{(x-3)(5-x)}+\frac{1}{2} \sin ^{-1}(x-4)+C$
D
None of these

✓

Answer

Correct option: B.

$\frac{1}{2}(x-4) \sqrt{(x-3)(5-x)}+\frac{1}{2} \sin ^{-1}(x-4)+C$

Let $ I=\int \sqrt{(x-3)(5-x)} d x=\int \sqrt{-x^2+8 x-15} d x$
$\Rightarrow I=\int \sqrt{-\left\{x^2-8 x+16-16+15\right\}} d x$
$\Rightarrow I=\int \sqrt{-\left\{(x-4)^2-1^2\right\}} d x=\int \sqrt{1^2-(x-4)^2} d x$
$\Rightarrow I=\frac{1}{2}(x-4) \sqrt{(x-3)(5-x)}+\frac{1}{2} \sin ^{-1}\left(\frac{x-4}{1}\right)+C$

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