a) \(\int {x{{\left( {2x - 3} \right)}^2}dx} = \int {x\left( {4{x^2} - 12x + 9} \right)dx} = \int {\left( {4{x^3} - 12{x^2} + 9x} \right)dx} \)
\( = 4\int {{x^3}dx} - 12\int {{x^2}dx} + 9\int {xdx} = 4.\frac{{{x^4}}}{4} - 12.\frac{{{x^3}}}{3} + 9.\frac{{{x^2}}}{2} + C = {x^4} - 4{x^3} + \frac{9}{2}{x^2} + C\)
b) \(\int {{{\sin }^2}\frac{x}{2}dx} = \int {\frac{{1 - \cos x}}{2}dx = \frac{1}{2}\int {dx} - \frac{1}{2}\int {\cos xdx} = \frac{1}{2}x - \frac{1}{2}\sin x + C} \)
c) \(\int {{{\tan }^2}xdx} = \int {\left( {\frac{1}{{{{\cos }^2}x}} - 1} \right)dx = \int {\frac{1}{{{{\cos }^2}x}}dx} - \int {dx} = \tan x - x + C} \)
d) \(\int {{2^{3x}}{{.3}^x}} dx = \int {{{\left( {{2^3}} \right)}^x}{{.3}^x}dx} = \int {{8^x}{{.3}^x}dx} = \int {{{24}^x}dx} = \frac{{{{24}^x}}}{{\ln 24}} + C\)