What is `int_(0)^(1) (tan^(-1))/(1+x^(2))` equal to ?

Options

A. `pi/4`
B. `pi/8`
C. `(pi^(2))/(8)`
D. `(pi^(3))/(32)`

Answer

D

Explanation

Let `I = underset(0)overset(1)(O)(tan^(-1))/(1+x^(2)) dx`
Put `tan^(-1) x = t` ltbtgt `(1)/(1+x^(2)) dx = dt`
`x = 0, rArr t = 0`
`x = 1, rArr t = pi//4`
`:. I = int_(0)^(pi//4) = (pi^(2))/(32)`