$ \int \frac{(x + x^{2/3} + x^{1/6})}{x(1 + x^{1/3})} dx $ is equal to

Q: $\large \int \frac{(x + x^{2/3} + x^{1/6})}{x(1 + x^{1/3})} dx $ is equal to

(A) $\large \frac{3}{2}x^{2/3} + 6 tan^{-1}(x^{1/6}) + c $

(B) $\large \frac{3}{2}x^{2/3} – 6 tan^{-1}(x^{1/6}) + c $

(D) none of these

Sol: Substituting x = z⁶, dx = 6 z⁵dz , we have

$\large I = \int \frac{6 z^5 (z^6 + z^4 + z)}{z^6 (1+z^2)}dz $

$\large = \int \frac{6 (z^5 + z^3 + 1)}{(1+z^2)}dz $

$\large = \int 6 z^3 dz + \int (\frac{6 dz}{z^2 + 1}) $

$\large = 6\frac{z^4}{4} + 6 tan^{-1}z + c $

$\large \frac{3}{2}x^{2/3} + 6 tan^{-1}(x^{1/6}) + c $

Hence (A) is the correct answer.