The value of the integral ∫ (x)/((1 + x)(1 + x^2))dx, x ∈ [0, ∞] A. π/2 B. π/2 C. π/6 D. π/3

Question

The value of the integral \(\int\limits_{0}^{\infty}\cfrac{\text x}{(1+\text x)(1+\text x^2)}d\text x \) is

A. \(\cfrac{\pi}2\) 

B. \(\cfrac{\pi}4\)

C. \(\cfrac{\pi}6\)

D.\(\cfrac{\pi}3\)

Answer 1

Correct option is B. \(\cfrac\pi4\)

Let, x = tan t

Differentiating both side with respect to t

\(\cfrac{dx}{dt}=sec^2t\)

⇒ dx = sec²t dt

At x = 0, t = 0

At x = ∞, t = π\2

By using the king rule

On adding eq(1) and eq(2)