Упр.1034 ГДЗ Алимов 10-11 класс (Алгебра)

Вычислить интеграл:

1. $$\int_{-1}^{2} 2\,dx$$;
2. $$\int_{-2}^{2} (3-x)\,dx$$;
3. $$\int_{1}^{3} (x^2-2x)\,dx$$;
4. $$\int_{-1}^{1} (2x-3x^2)\,dx$$;
5. $$\int_{1}^{8} \sqrt[3]{x}\,dx$$;
6. $$\int_{1}^{2} \frac{dx}{x^3}$$;
7. $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x\,dx$$.

1. $$\int_{-1}^{2} 2\,dx = 2x\Big|_{-1}^{2} = 2\cdot 2 — 2\cdot(-1) = 4+2=6.$$

   Ответ: $$6$$

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

   $$\left(3\cdot 2-\frac{2^2}{2}\right)-\left(3\cdot(-2)-\frac{(-2)^2}{2}\right)= (6-2)-(-6-2)=12.$$

   Ответ: $$12$$

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

   $$\left(\frac{27}{3}-9\right)-\left(\frac{1}{3}-1\right)=0-\left(-\frac{2}{3}\right)=\frac{2}{3}.$$

   Ответ: $$\frac{2}{3}$$

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

   $$\left(1-1\right)-\left(1-(-1)\right)=0-2=-2.$$

   Ответ: $$-2$$

5. $$\int_{1}^{8} \sqrt[3]{x}\,dx = \int_{1}^{8} x^{1/3}\,dx = \frac{3}{4}x^{4/3}\Big|_{1}^{8}.$$

   $$\frac{3}{4}\left(8^{4/3}-1^{4/3}\right)=\frac{3}{4}(16-1)=\frac{45}{4}=11\frac{1}{4}.$$

   Ответ: $$11\frac{1}{4}$$

6. $$\int_{1}^{2} \frac{dx}{x^3}=\int_{1}^{2} x^{-3}\,dx=-\frac{1}{2x^2}\Big|_{1}^{2}.$$

   $$-\frac{1}{2\cdot 2^2}+\frac{1}{2\cdot 1^2}=-\frac{1}{8}+\frac{1}{2}=\frac{3}{8}.$$

   Ответ: $$\frac{3}{8}$$

7. $$\int_{-\pi/2}^{\pi/2} \cos x\,dx = \sin x\Big|_{-\pi/2}^{\pi/2}.$$

   $$\sin\frac{\pi}{2}-\sin\left(-\frac{\pi}{2}\right)=1-(-1)=2.$$

   Ответ: $$2$$