IIT JEE , AIEEE Forum - Mathematics , Differential Calculus - INTEGRATE THE FUNCTION- f(x)= xf(cosx) FROM 0 to 2 pie ?

feynmann

Giving u a simpler soln .

Let g'( cosx ) = f ( cos x) ( It can always be written )

so we have ,

_{[0 ]}

^{[ 2pi]} f ( cos x ) dx = g ( cos ( 2pi ) ) - g ( cos ( 0 )) = 0 ...................(1 )

Now let I = given integral

change x to 2 pi - x. so the integral becomes

I =

_{[ 0]}

^{[2pi ]} 2pi f ( cos x ) dx - I

( since f( cos ( 2pi -x ) )= f ( cos x ) )

So we get I = pi

f ( cos x ) dx = 0 ( by 1 )