If -0-2 f sin 2 x sin x dx -2-4-0-4 f cos 2 x cos x dx then the value of - must be

I = ∫_0^π/2 f(sin 2x) sin x dx . . . (1) = ∫_0^π/2 f(sin 2 ( π/2 x ))sin (π/2 x )dx = ∫_0^π/2 f(sin 2x) cos x dx 2I = ∫_0^π/2 f(sin 2x)(cos x + sin x)dx 2I = ...

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Byju's Answer

Standard XII

Mathematics

Integration of Trigonometric Functions

If ∫0π/2 f si...

Question

If $\int_{0}^{\frac{π}{2}} f (s i n 2 x) s i n x d x = \frac{λ \sqrt{2}}{4} \int_{0}^{\frac{π}{4}} f (c o s 2 x) c o s x d x$ then the value of $λ$ must be ___

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Solution

$I = \int_{0}^{\frac{π}{2}} f (s i n 2 x) s i n x d x$ . . . (1)

$= \int_{0}^{\frac{π}{2}} f (s i n 2 (\frac{π}{2} - x)) s i n (\frac{π}{2} - x) d x$

$= \int_{0}^{\frac{π}{2}} f (s i n 2 x) c o s x d x$

$2 I = \int_{0}^{\frac{π}{2}} f (s i n 2 x) (c o s x + s i n x) d x$

$2 I = \sqrt{2} \int_{0}^{\frac{π}{2}} f (s i n 2 x) c o s (x - \frac{π}{4}) d x$

$2 I = \sqrt{2} \int_{- \frac{π}{4}}^{\frac{π}{4}} f (s i n 2 (x + \frac{π}{4})) c o s x d x$

$2 I = \sqrt{2} \int_{- \frac{π}{4}}^{\frac{π}{4}} f (c o s 2 x) c o s x d x$

$2 I = 2 \sqrt{2} \int_{0}^{\frac{π}{4}} f (c o s 2 x) c o s x d x$

$∴ I = \sqrt{2} \int_{0}^{\frac{π}{4}} f (c o s 2 x) c o s x d x$

$λ = 4$

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Similar questions

Q. If

\int_{0}^{π / 2} f (sin 2 x) sin x d x = A \frac{\sqrt{2}}{9} \int_{0}^{π / 4} f (cos 2 x) cos x d x

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π / 2 \int 0 f (sin 2 x) sin x d x = \sqrt{2} π / 4 \int 0 f (cos 2 x) c o s x d x

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and

I_{2} = \int_{0}^{\frac{π}{4}} f (cos 2 x) cos x d x,

then

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If ∫π20 f(sin 2x) sin x dx=λ√24 ∫π40 f(cos 2x) cos x dx then the value of λ must be ___

If $\int_{0}^{\frac{π}{2}} f (s i n 2 x) s i n x d x = \frac{λ \sqrt{2}}{4} \int_{0}^{\frac{π}{4}} f (c o s 2 x) c o s x d x$ then the value of $λ$ must be ___