Correct answer is (b) xtan^-1 (x) – ^1⁄2 ln(1 + x^2)
To explain: Add constant automatically
Given, ∫tan^-1(x)dx
Putting, x = tan(y),
We get, dy = sec^2(y)dy,
∫ysec^2(y)dy
By integration by parts,
ytan(y) – log(sec(y)) = xtan^-1 (x) – ^1⁄2 ln(1 + x^2).