Explanation

(a)
Since, $\int_{1}^{a}{\lbrack x\rbrack f^{'}(x)dx = \int_{1}^{2}{f^{'}(x)dx}}$
$+ \int_{2}^{3}{2f^{'}(x)dx + ... + \int_{\lbrack a\rbrack}^{a}{\lbrack a\rbrack f^{'}(x)dx}}$
$= \lbrack f{(x)\rbrack}_{1}^{2} + 2\left\lbrack f(x) \right\rbrack_{2}^{3} + ... + \lbrack a\rbrack\left\lbrack f(x) \right\rbrack_{\lbrack a\rbrack}^{a}$
$= f(2) - f(1) + 2f(3) - 2f(2) + ...$
$+ \lbrack a\rbrack f(a) - \lbrack a\rbrack f(\lbrack a\rbrack)$
$= \lbrack a\rbrack f(a) - \{ f(1) + f(2) + ... + f\left( \lbrack a\rbrack \right)\}$