IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

man111

(1)Evaluate _{[0 ]}^[infinity]sinkx(cosx)^k/x, where (k belong to I⁺).

plz gave the explanation of this.

(2)how can we solve f(x), f(y)

if the expression of the form f(x+y)+f(x-y)=2f(x).f(y)

sboosy

f(x+y)+f(x-y)=2f(x)f(y)

put y= -x

f(0)+f(2x)=2f(x)f(-x) -----1

putx= -y

f(0)+f(-2y)=2f(-y)f(y) -----2

since function is variable immaterial

subtracting 1 from 2 we get

f(2x)-f(-2x)=0

which means f is even

now in 1 put x=0

which gives f(0)=either 0 or 1.

case1 (f(0)=0)

put y=0 in main equation

f(x)+f(x)=2f(x)f(0)

2f(x)[1-f(0)]=0

since f(0)=0

f(x)=0 thus it is a constant function

case2 (f(0)=1)

there r some possibilities where each value is different

but one immediate conclusion is f assuming a constant value

of 1

Another way:

I think (correct me if i am wrong)

that the fn given is such that it can assume some constant value k

then according to question

k+k=2k*k

which gives value of k as either 1 or 0

f(x) can be a constant function taking wither 0 or 1