Tính các tích phân sau:

13/16

a) \(\int\limits_1^4 {\left( {{x^3} - 2\sqrt x } \right)dx} \);

b) \(\int\limits_0^{\frac{\pi }{2}} {\left( {\cos x - \sin x} \right)dx} \);

c) \(\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{4}} {\frac{{dx}}{{{{\sin }^2}x}}} \);

d) \(\int\limits_1^{16} {\frac{{x - 1}}{{\sqrt x }}dx} \).

Nhập câu trả lời ở đây

0/3000 ký tự

Giải thích

a) \(\int\limits_1^4 {\left( {{x^3} - 2\sqrt x } \right)dx} \)\( = \int\limits_1^4 {{x^3}dx} - 2\int\limits_1^4 {{x^{\frac{1}{2}}}dx} \)\( = \left. {\left( {\frac{{{x^4}}}{4} - \frac{4}{3}{x^{\frac{3}{2}}}} \right)} \right|_1^4\)\( = \frac{{160}}{3} + \frac{{13}}{{12}} = \frac{{653}}{{12}}\).

b) \(\int\limits_0^{\frac{\pi }{2}} {\left( {\cos x - \sin x} \right)dx} \)\( = \int\limits_0^{\frac{\pi }{2}} {\cos xdx} - \int\limits_0^{\frac{\pi }{2}} {\sin xdx} \)\( = \left. {\left( {\sin x + \cos x} \right)} \right|_0^{\frac{\pi }{2}} = 1 - 1 = 0\).

c) \(\int\limits_{\frac{\pi }{6}}^{\frac{\pi }{4}} {\frac{{dx}}{{{{\sin }^2}x}}} = \left. { - \cot x} \right|_{\frac{\pi }{6}}^{\frac{\pi }{4}} = - 1 + \sqrt 3 \).

d) \(\int\limits_1^{16} {\frac{{x - 1}}{{\sqrt x }}dx} \)\( = \int\limits_1^{16} {\sqrt x dx} - \int\limits_1^{16} {\frac{1}{{\sqrt x }}dx} \)\( = \int\limits_1^{16} {{x^{\frac{1}{2}}}dx} - \int\limits_1^{16} {{x^{ - \frac{1}{2}}}dx} \)\( = \left. {\left( {\frac{2}{3}{x^{\frac{3}{2}}} - 2{x^{\frac{1}{2}}}} \right)} \right|_1^{16}\)\( = \frac{{104}}{3} + \frac{4}{3} = 36\).