(In reply to
Puzzle Answer by K Sengupta)
At the outset, substituting x = sin t, we have:
f(sin t) + f(cos t) = 2 ..........(i)
Now,
I = Integral (0 to 1) {f(x)/V(1-x^2)} dx (say)
Substituting x= cos t, we have:
I = Integral (pi/2 to 0) {f(cos t)/sin t} * (-sin t) dt
= Integral (0 to pi/2) f(cos t) dt ..........(ii)
Again, substituting x= sin t, we have:
I = Integral (0 to pi/2) *{f(sin t)/cos t} * cos t dt
= Integral (0 to pi/2)* f(sin t) dt ........(iii)
Adding (ii) and (iii), we obtain:
2I = Integral (0 to pi/2){f(sin t) + f(cos t)} dt
= Integral {0 to pi/2}*(2) dt {from (i)}
= 2 * Integral(0 to pi/2) dt
= 2*(pi/2)
=> I = pi/2
Consequently, the required definite Integral evaluates to pi/2
Edited on July 22, 2022, 1:32 am
Explanation to Puzzle Answer
