Упр.1006 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\int_{-3}^{2}(2x-3)\,dx$$
2. $$\int_{-2}^{-1}(5-4x)\,dx$$
3. $$\int_{-1}^{2}(1-3x^2)\,dx$$
4. $$\int_{-1}^{1}(x^2+1)\,dx$$
5. $$\int_{0}^{2}(3x^2-4x+5)\,dx$$

Вычислим интегралы по формуле Ньютона–Лейбница:

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

   $$=(4-6)-\bigl(9+9\bigr)=-2-18=-20.$$

2. $$\int\limits_{-2}^{-1}(5-4x)\,dx=\left(5x-2x^2\right)\Big|_{-2}^{-1}$$

   $$=(-5-2)-(-10-8)=-7-(-18)=11.$$

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

   $$=(2-8)-\bigl(-1-(-1)\bigr)=-6-0=-6.$$

4. $$\int\limits_{-1}^{1}(x^2+1)\,dx=\left(\frac{x^3}{3}+x\right)\Big|_{-1}^{1}$$

   $$=\left(\frac13+1\right)-\left(-\frac13-1\right)=\frac43+\frac43=\frac83=2\frac23.$$

5. $$\int\limits_{0}^{2}(3x^2-4x+5)\,dx=\left(x^3-2x^2+5x\right)\Big|_{0}^{2}$$

   $$=(8-8+10)-0=10.$$

Ответ: 1) $$-20$$; 2) $$11$$; 3) $$-6$$; 4) $$\frac83$$; 5) $$10$$.