If f : be a continuous function satisfying 0 / 2 f ( sin 2 x ) sin x ? d x + 0 / 4 f ( cos 2 x ) cos x ? d x = 0 , then the value of is [2023]

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Solution

Q.
If $f : ℝ \to ℝ$ be a continuous function satisfying $\int_{0}^{π / 2} f (\sin 2 x) \sin x d x + α \int_{0}^{π / 4} f (\cos 2 x) \cos x d x = 0,$ then the value of $α$ is [2023]

1 $- \sqrt{2}$

2 $\sqrt{3}$

3 $- \sqrt{3}$

4 $\sqrt{2}$

Ans.
(1)

$I = \int_{0}^{π / 4} f (\sin 2 x) \sin x d x + \int_{π / 4}^{π / 2} f (\sin 2 x) \sin x d x + α \int_{0}^{π / 4} f (\cos 2 x) \cos x d x = 0$

$Apply \int_{0}^{a} f (x) d x = \int_{0}^{a} f (a - x) d x in the first part and put x - \frac{π}{4} = t in the second part, we get$

$I = \int_{0}^{π / 4} f (\cos 2 x) \sin (\frac{π}{4} - x) d x + \int_{0}^{π / 4} f (\cos 2 t) \sin (\frac{π}{4} + t) d t + α \int_{0}^{π / 4} f (\cos 2 x) \cos x d x = 0$

$= \int_{0}^{π / 4} f (\cos 2 x) [2 \sin \frac{π}{4} \cdot \cos x + α \cos x] d x = 0$

$(α + \sqrt{2}) \int_{0}^{π / 4} f (\cos 2 x) \cos x d x = 0 ∴ α = - \sqrt{2}$

$[∵ In (0, \frac{π}{4}), f (\cos (2 x)) and \cos x are not zero.]$

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