Explanation

(c)
(A)$\int_{0}^{1}e^{x}dx = \left\lbrack e^{x} \right\rbrack_{0}^{1} = e - 1$
(B)$\int_{0}^{1}2^{x}dx = \left\lbrack \frac{2^{x}}{\log_{e}2} \right\rbrack_{0}^{1} = \frac{1}{\log 2}.\left( 2 - 2^{0} \right) = \frac{1}{\log 2}$
(C )$\int_{0}^{1}\sqrt{x}dx = \left\lbrack \frac{x^{\frac{3}{2}}}{\frac{3}{2}} \right\rbrack_{0}^{1} = \frac{2}{3}$
(D)$\int_{0}^{1}xdx = \left\lbrack \frac{x^{2}}{2} \right\rbrack_{0}^{1} = \frac{1}{2}$