Tính các tích phân sau: a) tích phân từ -2 đến 4 của (x+1)(x-1) dx ; b)tích phân từ 1 đến 2 của x^2 -2x +1/ x dx

a) \(\int\limits_{ - 2}^4 {\left( {x + 1} \right)\left( {x - 1} \right)dx} \)\( = \int\limits_{ - 2}^4 {\left( {{x^2} - 1} \right)dx} \)\( = \int\limits_{ - 2}^4 {{x^2}dx} - \int\limits_{ - 2}^4 {dx} \)\( = \left. {\left( {\frac{{{x^3}}}{3} - x} \right)} \right|_{ - 2}^4\)\( = \frac{{52}}{3} + \frac{2}{3} = \frac{{54}}{3} = 18\).

b) \(\int\limits_1^2 {\frac{{{x^2} - 2x + 1}}{x}dx} \)\( = \int\limits_1^2 {\left( {x + \frac{1}{x} - 2} \right)dx} \)\( = \left. {\left( {\frac{{{x^2}}}{2} + \ln \left| x \right| - 2x} \right)} \right|_1^2\)

\( = - 2 + \ln 2 + \frac{3}{2} - \ln 1 = \ln 2 - \frac{1}{2}\).

c) \(\int\limits_0^{\frac{\pi }{2}} {\left( {3\sin x - 2} \right)dx} \)\( = 3\int\limits_0^{\frac{\pi }{2}} {\sin xdx} - 2\int\limits_0^{\frac{\pi }{2}} {dx} \)\( = \left. {\left( { - 3\cos x - 2x} \right)} \right|_0^{\frac{\pi }{2}}\)\( = - \pi + 3\).

d) \(\int\limits_0^{\frac{\pi }{2}} {\frac{{{{\sin }^2}x}}{{1 + \cos x}}dx} \)\( = \int\limits_0^{\frac{\pi }{2}} {\frac{{1 - {{\cos }^2}x}}{{1 + \cos x}}dx} \)\( = \int\limits_0^{\frac{\pi }{2}} {\left( {1 - \cos x} \right)dx} \)\( = \int\limits_0^{\frac{\pi }{2}} {dx} - \int\limits_0^{\frac{\pi }{2}} {\cos xdx} \)

\( = \left. {\left( {x - \sin x} \right)} \right|_0^{\frac{\pi }{2}}\)\( = \frac{\pi }{2} - 1\).