Question

The value of

i log (x –i) + i²p + i³ log (x + i) + i⁴ (2 tan^–1x) where x > 0 and i = $\sqrt{- 1}$is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (A)

0

Answer 2

I log $\frac{x - i}{x + i}$– p + 2 tan^–1 x = z (let)

\ log $\left( \frac{x + i}{x–i} \right)$ = i (z + p –2 tan^–1 x)

or $\frac{x + i}{x–i}$= e^iq where q = z + p – 2 tan^–1 x

x + i = x cos q + i x sin q + i sin q – i cos q

̃ x = cot $\frac{\theta}{2}$ ̃ q = 2 cot^–1x

or z + p – 2 tan^–1 x = 2 cot^–1 x ̃ z = 0