Tính: a) tích phân 0 pi/4 sin^x / 2; b)tích phân 0 1 {3x - 4x^3dx - tích phân 1 2 4x^3 - 3x;

29/45

Tính:

a) \(\int\limits_0^{\frac{\pi }{4}} {{{\sin }^2}\frac{x}{2}dx} \);

b) \(\int\limits_0^1 {\left( {3x - 4{x^3}} \right)dx - \int\limits_1^2 {\left( {4{x^3} - 3x} \right)dx} } \);

c) \(\int\limits_0^6 {\left( {\left| {2x - 2} \right| + 4{x^2}} \right)dx} \).

Nhập câu trả lời ở đây

0/3000 ký tự

Giải thích

a) \(\int\limits_0^{\frac{\pi }{4}} {{{\sin }^2}\frac{x}{2}dx} \) = \(\int\limits_0^{\frac{\pi }{4}} {\left( {\frac{{1 - \cos x}}{2}} \right)dx} = \int\limits_0^{\frac{\pi }{4}} {\frac{1}{2}dx} - \int\limits_0^{\frac{\pi }{4}} {\frac{{\cos x}}{2}dx} \) = \(\left. {\frac{1}{2}x} \right|_0^{\frac{\pi }{4}} - \left. {\frac{{\sin x}}{2}} \right|_0^{\frac{\pi }{4}}\) = \(\frac{\pi }{8} - \frac{{\sqrt 2 }}{4}\).

b) \(\int\limits_0^1 {\left( {3x - 4{x^3}} \right)dx - \int\limits_1^2 {\left( {4{x^3} - 3x} \right)dx} } \) = \(\left. {\left( {\frac{3}{2}{x^2} - {x^4}} \right)} \right|_0^1 - \left. {\left( {{x^4} - \frac{3}{2}{x^2}} \right)} \right|_1^2\)

= \(\left( {\frac{3}{2}{{.1}^2} - {1^4} - \frac{3}{2}{{.0}^2} + {0^4}} \right)\) − \(\left( {{2^4} - \frac{3}{2}{{.2}^2} - {1^4} + \frac{3}{2}{{.1}^2}} \right)\)

= 11.

c) \(\int\limits_0^6 {\left( {\left| {2x - 2} \right| + 4{x^2}} \right)dx} \) = \(\int\limits_0^1 {\left( {\left| {2x - 2} \right| + 4{x^2}} \right)dx} + \int\limits_0^6 {\left( {\left| {2x - 2} \right| + 4{x^2}} \right)dx} \)

= \(\int\limits_0^1 {\left( {2 - 2x + 4{x^2}} \right)dx} + \int\limits_0^6 {\left( {2x - 2 + 4{x^2}} \right)dx} \)

= \(\left. {\left( {2x - {x^2} + \frac{4}{3}{x^3}} \right)} \right|_0^1 - \left. {\left( {2x - {x^2} + \frac{4}{3}{x^3}} \right)} \right|_1^6\)

= 314.