Tìm: a) nguyên hàm x( 2x-3)^2 dx ; b)nguyên hàm sin^2 x/2 dx ;

a) \(\int {x{{\left( {2x - 3} \right)}^2}dx} \)\( = \int {x\left( {4{x^2} - 12x + 9} \right)dx} \)\( = \int {\left( {4{x^3} - 12{x^2} + 9x} \right)dx} \)

\( = {x^4} - 4{x^3} + \frac{9}{2}{x^2} + C\).

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) \(\int {{2^{3x}}{{.3}^x}dx} \)\( = \int {{8^x}{{.3}^x}dx} \)\( = \int {{{24}^x}dx} \)\( = \frac{{{{24}^x}}}{{\ln 24}} + C\).