Страница 315 номер 1034, ГДЗ по алгебре за 10-11 класс Алимов - ответы и решение
Вычислить интеграл: 1) \(\int\limits_{-1}^{2} 2\,dx\); 2) \(\int\limits_{-2}^{2} (3 - x)\,dx\); 3) \(\int\limits_{1}^{3} (x^2 - 2x)\,dx\); 4) \(\int\limits_{-1}^{1} (2x - 3x^2)\,dx\); 5) \(\int\limits_{1}^{8} \sqrt[3]{x}\,dx\); 6) \(\int\limits_{1}^{2} \frac{dx}{x^3}\); 7) \(\int\limits_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x\,dx\).
1) \(\int_{-1}^{2}{2dx} = \left. \ 2x \right|_{-1}^{2} = 4 + 2 = 6\);
2) \(\int_{-2}^{2}{(3 - x)dx} = \left. \ \left( 3x - \frac{x^{2}}{2} \right) \right|_{-2}^{2} = 6 - 2 + 6 + 2 = 12\);
3) \(\int_{1}^{3}{\left( x^{2} - 2x \right)dx} = \left. \ \left( \frac{x^{3}}{3} - x^{2} \right) \right|_{1}^{3} = 9 - 9 - \frac{1}{3} + 1 = \frac{2}{3}\);
4) \(\int_{-1}^{1}{\left( 2x - 3x^{2} \right)dx} = \left. \ \left( x^{2} - x^{3} \right) \right|_{-1}^{1} = 1 - 1 - 1 - 1 = - 2\);
5) \(\int_{1}^{8}{\sqrt[3]{x}dx} = \left. \ \frac{3}{4}x^{\frac{4}{3}} \right|_{1}^{8} = \frac{3}{4}(16 - 1) = \frac{45}{4} = 11\frac{1}{4}\);
6) \(\int_{1}^{2}\frac{dx}{x^{3}} = \left. \ - \frac{1}{2x^{2}} \right|_{1}^{2} = - \frac{1}{8} + \frac{1}{2} = \frac{3}{8}\);
7) \(\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\cos xdx} = \left. \ \sin x \right|_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = 1 + 1 = 2\).
