1correct !
original problem source : http://targetiit.com/iit-jee-forum/posts/challenge-1199.html
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)}
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5 Answers
1put\, \: ( 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
1
11definite integral first property:
0∫kf(x)=0∫kf(k-x)
I=0∫pie/3sin(x-(pie/6))........... →eq1
if we will apply first property
I would come out to be
I=-0∫pie/3sin(x-(pie/3))........same as above →eq2
adding eq1 and 2
we get
2I=0
I=0