Упр.1007 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\int_{0}^{4}\left(x-3\sqrt{x}\right)\,dx$$
2. $$\int_{1}^{9}\left(2x-\frac{3}{\sqrt{x}}\right)\,dx$$
3. $$\int_{0}^{2}e^{3x}\,dx$$
4. $$\int_{1}^{3}2e^{2x}\,dx$$

1) $$\int\limits_{0}^{4}(x-3\sqrt{x})\,dx=\left(\frac{x^2}{2}-2x\sqrt{x}\right)\Bigg|_{0}^{4}$$

$$\left(\frac{4^2}{2}-2\cdot4\sqrt{4}\right)-\left(\frac{0^2}{2}-2\cdot0\sqrt{0}\right)=8-16=-8$$

Ответ: $$-8$$

2) $$\int\limits_{1}^{9}\left(2x-\frac{3}{\sqrt{x}}\right)\,dx=\left(x^2-6\sqrt{x}\right)\Bigg|_{1}^{9}$$

$$\left(9^2-6\sqrt{9}\right)-\left(1^2-6\sqrt{1}\right)=81-18-1+6=68$$

Ответ: $$68$$

3) $$\int\limits_{0}^{2}e^{3x}\,dx=\frac{1}{3}e^{3x}\Bigg|_{0}^{2}$$

$$\frac{1}{3}\left(e^6-e^0\right)=\frac{e^6-1}{3}$$

Ответ: $$\frac{e^6-1}{3}$$

4) $$\int\limits_{1}^{3}2e^{2x}\,dx=e^{2x}\Bigg|_{1}^{3}$$

$$e^6-e^2$$

Ответ: $$e^6-e^2$$