Упр.49.1 ГДЗ Мордкович 10-11 класс (Алгебра)

a) интеграл(-2/3 1) x^3 dx;
6) интеграл(1 3) dx / x^2;
в) интеграл(-1 2) x^4 dx;
г) интеграл(4 9) dx / корень(x).

1. $$\int_{-\frac{2}{3}}^{1} x^3\,dx=\left.\frac{x^4}{4}\right|_{-\frac{2}{3}}^{1}=\frac{1^4}{4}-\frac{\left(-\frac{2}{3}\right)^4}{4}=\frac{1}{4}-\frac{16}{324}=\frac{81-16}{324}=\frac{65}{324}.$$

2. $$\int_{1}^{3}\frac{dx}{x^2}=\int_{1}^{3}x^{-2}\,dx=\left.-x^{-1}\right|_{1}^{3}=-\frac{1}{3}-\left(-1\right)=\frac{2}{3}.$$

3. $$\int_{-1}^{2} x^4\,dx=\left.\frac{x^5}{5}\right|_{-1}^{2}=\frac{2^5}{5}-\frac{(-1)^5}{5}=\frac{32+1}{5}=\frac{33}{5}.$$

4. $$\int_{4}^{9}\frac{dx}{\sqrt{x}}=\int_{4}^{9}x^{-\frac12}\,dx=\left.2x^{\frac12}\right|_{4}^{9}=2\sqrt{9}-2\sqrt{4}=2(3-2)=2.$$

Ответ

а) $$\frac{65}{324}$$; б) $$\frac{2}{3}$$; в) $$\frac{33}{5}$$; г) $$2$$.