IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

kbanagar

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^[^] [f(x)+f(-x)] [g(x)+g(-x)]dx
upper limit= +pie / 2
lower limit= -pie / 2
the options as follows.........
a] pie
b] 1
c] -1
d] zero........{crt answer}
pls explain in detail...............

hsbhatt

Let I = _-/2

^/2 (f(x)+g(x)) (f(-x)+g(-x)) dx

let t = -x. Then dt = -dx

Then we get I = - _-/2^/2 (f(t)+g(t)) (f(-t)+g(-t)) dt = -_-/2^/2 (f(x)+g(x)) (f(-x)+g(-x)) dt = -I

Hence I = 0.

sboosy

bhattji i think u didnt look at the question properly
the answer is going to b 0 only if either one of the functions is odd
bcos then f(-x)= - f(x) and similarly g(-x) = - g(x)
in this case answer is 0

now is both are even functions then given reduces to
integral 4 f(x) g (x)
this may b non zero

konichiwa2x

yeah, i think he accidently forgot to change the limits because

Case 1 : One or both of the functions are odd, Then obviosully the integral is 0 since

Case 2: Both are even. Then, the integral becomes

since

as pointed out by sboosy.

hsbhatt

Over lunch I thought I was missing something because with cosx I was getting a contradictory result. Thanx for pointing it out.

Lesson: Never do probs between jobs

krishna.gopal

Perfect answer sboosy and konichiwa. well done

kbanagar

the answer given is zero......................
n not the as said by sboosy.........
let me edit the question i'll giv the options to.....
do xplain in detail............

hsbhatt

There must have been a misprint. The integrand should be (f(x) - f(-x)) (g(x)+g(-x)) or (f(x) + f(-x)) (g(x)-g(-x)). In short, it should be an odd function.

kbanagar

but i've posted the thing correctly........
i.e (f(x) + f(-x)) (g(x)-g(-x))
the only xplanation is abt the question being given is a odd function.........

hsbhatt

There you go man, look at your original post you have said it is (f(x)+f(-x)) (g(x)+g(-x))

Well if the integrand is h(x) = (f(x)+f(-x)) (g(x)-g(-x))

Then h(-x) = (f(-x)+f(x)) (g(-x)-g(x))

= -(g(x)-g(-x)) (f(x)+f(-x))

= -h(x)

Thus it is established that h(x) is an odd function.

kbanagar

oh..........thanx a lot hsbhat...........

sboosy

kbanagar be more careful to provide all givens about the question next time around

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