Explanation

(a)
Let $I = \int_{0}^{\lambda}\frac{y\ dy}{\sqrt{y + \lambda}}$
$= \int_{0}^{\lambda}\left\lbrack \sqrt{y + \lambda} - \frac{\lambda}{\sqrt{y + \lambda}} \right\rbrack dy$
$= \left\lbrack \frac{(y + \lambda)^{\frac{3}{2}}}{\frac{3}{2}} \right\rbrack_{0}^{\lambda} - \left\lbrack \frac{\lambda\sqrt{y + \lambda}}{\frac{1}{2}} \right\rbrack_{0}^{\lambda}$
$= \frac{2}{3}\left\lbrack (2\lambda)^{\frac{3}{2}} - \lambda^{\frac{3}{2}} \right\rbrack - 2\lambda\left\lbrack (2\lambda)^{\frac{1}{2}} - (\lambda)^{\frac{1}{2}} \right\rbrack$
$= 2\lambda\sqrt{\lambda}\left\lbrack \frac{2\sqrt{2} - 1}{3} - \left( \sqrt{2} - 1 \right) \right\rbrack$
$= \frac{2}{3}\ \lambda\sqrt{\lambda}(2 - \sqrt{2})$