Question

If f(x) = ∫log[(1 – t)/(1 + t)]dt for t ∈ [0, x] then f(1/2) – f[(– 1)/2] is equals to.......... 

(a) 0

(b) (1/2)

(c) [(– 1)/2]

(d) 1

Answer 1

The correct option (a) 0

Explanation:

f(x) = ^x∫₀log[(1 – t)/(1 + t)]dt 

f(1/2) – f[(– 1)/2] 

= ^(1/2)∫₀log[(1 – t)/(1 + t)]dt – ^[(–1)/2]∫₀log[(1 – t)/(1 + t)]dt 

= ^(1/2)∫₀log[(1 – t)/(1 + t)]dt + ⁰∫_[(–1)/2]log[(1 – t)/(1 + t)]dt 

= ^(1/2)∫_[(–1)/2]log[(1 – t)/(1 + t)]dt 

Now log[(1 – t)/(1 + t)] is an odd function of t 

∴ f(1/2) – f[(– 1)/2] = 0.