Упр.1008 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\int_{-2}^{1} x(x+3)(2x-1)\,dx$$
2. $$\int_{-1}^{0} (x+1)(x^2-2)\,dx$$
3. $$\int_{1}^{2} \left(x+\frac{1}{x}\right)^2\,dx$$
4. $$\int_{-2}^{-1} \frac{4}{x^2}\left(1-\frac{2}{x}\right)\,dx$$

1)

$$\int_{-2}^{1} x(x+3)(2x-1)\,dx = \int_{-2}^{1} (2x^3+5x^2-3x)\,dx$$
$$= \left(\frac{x^4}{2}+\frac{5x^3}{3}-\frac{3x^2}{2}\right)\Bigg|_{-2}^{1}$$
$$= \left(\frac12+\frac53-\frac32\right)-\left(\frac{16}{2}+\frac{5\cdot(-8)}{3}-\frac{3\cdot4}{2}\right) = -\frac{3}{2}+15=12$$

2)

$$\int_{-1}^{0} (x+1)(x^2-2)\,dx = \int_{-1}^{0} (x^3+x^2-2x-2)\,dx$$
$$= \left(\frac{x^4}{4}+\frac{x^3}{3}-x^2-2x\right)\Bigg|_{-1}^{0}$$
$$= 0-\left(\frac14-\frac13-1+2\right) = -\frac{11}{12}$$

3)

$$\int_{1}^{2}\left(x+\frac{1}{x}\right)^2 dx = \int_{1}^{2}\left(x^2+2+\frac{1}{x^2}\right)dx$$
$$= \left(\frac{x^3}{3}+2x-\frac{1}{x}\right)\Bigg|_{1}^{2}$$
$$= \left(\frac{8}{3}+4-\frac12\right)-\left(\frac13+2-1\right) = \frac{11}{6}+3=\frac{29}{6}=4\frac{5}{6}$$

4)

$$\int_{-2}^{-1}\frac{4}{x^2}\left(1-\frac{2}{x}\right)\,dx = \int_{-2}^{-1}\left(\frac{4}{x^2}-\frac{8}{x^3}\right)\,dx$$
$$= \left(-\frac{4}{x}+\frac{4}{x^2}\right)\Bigg|_{-2}^{-1}$$
$$= \left(4+4\right)-\left(2+1\right)=5$$

Ответ: 1) $$12$$; 2) $$-\frac{11}{12}$$; 3) $$4\frac{5}{6}$$; 4) $$5$$.