integral challenge for 2011 aspirants

JEE Advanced 2015 Preparation › Calculus questions › integral challenge for 2011 aspirants

find :

\int_{0}^{\frac{\pi}{3}}{\sin{\left( x-\frac{\pi}{6}}\right)\sin{\left( 2x-\frac{\pi}{3}}\right)\sin{\left( 3x-\frac{\pi}{2}}\right)\sin{\left( 4x-\frac{2\pi}{3}}\right)\sin{\left( 5x-\frac{5\pi}{6}}\right)}

#Mathematics #Calculus

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5 Answers

106

Asish Mahapatra ·2010-06-28 20:49:24

ANSWER : 0

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1

Sonne ·2010-06-28 20:59:39

not for those who have already cracked jee :P

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1

à¤œà¤¯ ·2010-06-28 21:02:51

put\, \: ( 3x -\frac{\pi }{2}) = t

3 dx = dt

at \: \: x= \pi /3 \rightarrow t=\frac{3.\pi }{3} -\frac{\pi }{2} = \frac{\pi }{2}

at x=0 t = -\frac{\pi }{2}

hence function is odd function and ans is 0

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1

Sonne ·2010-06-28 21:06:14

correct !
original problem source : http://targetiit.com/iit-jee-forum/posts/challenge-1199.html

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11

vaibhav sharma ·2010-06-28 22:21:47

definite integral first property:

₀âˆ«^kf(x)=₀âˆ«^kf(k-x)

I=₀âˆ«^pie/3sin(x-(pie/6))........... â†’eq1

if we will apply first property

I would come out to be

I=-₀âˆ«^pie/3sin(x-(pie/3))........same as above â†’eq2

adding eq1 and 2
we get
2I=0
I=0

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Your Answer

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° ∂ ∞ α β γ δ θ λ μ ν ξ π ρ σ φ ω ε ƒ κ τ υ Γ Δ Λ Σ Π Ω