Упр.371 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

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

1. $$\int_{-2}^{1} x(x+3)(2x-1)\,dx=\int_{-2}^{1}(x^2+3x)(2x-1)\,dx=\int_{-2}^{1}(2x^3+5x^2-3x)\,dx.$$

   $$\int_{-2}^{1}(2x^3+5x^2-3x)\,dx=\left(\frac{x^4}{2}+\frac{5x^3}{3}-\frac{3x^2}{2}\right)\Bigg|_{-2}^{1}=12.$$

2. $$\int_{1}^{2}\left(x+\frac{1}{x}\right)^2dx=\int_{1}^{2}\left(x^2+2+\frac{1}{x^2}\right)dx.$$

   $$\int_{1}^{2}\left(x^2+2+\frac{1}{x^2}\right)dx=\left(\frac{x^3}{3}+2x-\frac{1}{x}\right)\Bigg|_{1}^{2}=\frac{14}{3}-\frac{3}{2}=\frac{19}{6}.$$

3. $$\int_{-1}^{0}(x+1)(x^2-2)\,dx=\int_{-1}^{0}(x^3+x^2-2x-2)\,dx.$$

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

4. $$\int_{-2}^{-1}\frac{4}{x^2}\left(1-\frac{2}{x}\right)dx=\int_{-2}^{-1}\left(\frac{4}{x^2}-\frac{8}{x^3}\right)dx.$$

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

Ответ

1) $$12$$; 2) $$\frac{19}{6}$$; 3) $$-\frac{11}{12}$$; 4) $$5$$.