Номер 88, страница 19 - гдз по алгебре 11 класс дидактические материалы Мерзляк, Полонский

88. Вычислите интеграл:

1) $\int_{-1}^{3} (x + 2)dx;$

2) $\int_{0}^{5} (x^2 - 3x)dx;$

3) $\int_{-3}^{2} (x - 4)^2 dx;$

14) $\int_{8}^{27} \sqrt[3]{x}dx;$

4) $\int_{1}^{2} (5x - 9)^4 dx;$

15) $\int_{-1}^{1} \sqrt{1 - x}dx;$

5) $\int_{-2.5}^{-2} \frac{8dx}{(2x + 3)^3};$

16) $\int_{\frac{\pi}{3}}^{\frac{2\pi}{3}} \cos\left(\frac{\pi}{3} - 3x\right) dx;$

6) $\int_{1}^{9} \left(1 + \frac{2}{\sqrt{x}}\right) dx;$

17) $\int_{0}^{\ln 5} e^x dx;$

7) $\int_{-1}^{1} \frac{dx}{\sqrt{4x + 5}};$

18) $\int_{2}^{3} 5^x dx;$

8) $\int_{0}^{6} \frac{dx}{\sqrt{4 - \frac{x}{2}}};$

19) $\int_{-8}^{0} e^{-\frac{x}{8}} dx;$

9) $\int_{\frac{\pi}{6}}^{\pi} \cos x dx;$

20) $\int_{3}^{27} \frac{dx}{x};$

10) $\int_{\frac{\pi}{2}}^{\pi} \sin \frac{x}{2} dx;$

21) $\int_{e}^{e^5} \frac{3}{x} dx;$

11) $\int_{\frac{3\pi}{4}}^{\pi} \frac{dx}{\cos^2 x};$

22) $\int_{-3}^{-1} \left(\frac{4}{x} - x\right) dx;$

12) $\int_{\frac{\pi}{12}}^{\frac{\pi}{9}} \frac{dx}{\sin^2 3x};$

23) $\int_{0}^{5} \frac{dx}{7x + 5};$

13) $\int_{4}^{9} \sqrt{x}dx;$

24) $\int_{-2}^{0} \frac{dx}{3x - 2};$

1) $\int_{-1}^{3} (x + 2) dx = \left(\frac{x^2}{2} + 2x\right) \Big|_{-1}^{3} = \left(\frac{3^2}{2} + 2 \cdot 3\right) - \left(\frac{(-1)^2}{2} + 2 \cdot (-1)\right) = \left(\frac{9}{2} + 6\right) - \left(\frac{1}{2} - 2\right) = \frac{21}{2} - \left(-\frac{3}{2}\right) = \frac{24}{2} = 12$.
Ответ: $12$.

2) $\int_{0}^{5} (x^2 - 3x) dx = \left(\frac{x^3}{3} - \frac{3x^2}{2}\right) \Big|_{0}^{5} = \left(\frac{5^3}{3} - \frac{3 \cdot 5^2}{2}\right) - 0 = \frac{125}{3} - \frac{75}{2} = \frac{250 - 225}{6} = \frac{25}{6}$.
Ответ: $\frac{25}{6}$.

3) $\int_{-3}^{2} (x - 4)^2 dx = \left(\frac{(x-4)^3}{3}\right) \Big|_{-3}^{2} = \frac{(2-4)^3}{3} - \frac{(-3-4)^3}{3} = \frac{(-2)^3}{3} - \frac{(-7)^3}{3} = \frac{-8 - (-343)}{3} = \frac{335}{3}$.
Ответ: $\frac{335}{3}$.

4) $\int_{1}^{2} (5x - 9)^4 dx = \left(\frac{(5x-9)^5}{5 \cdot 5}\right) \Big|_{1}^{2} = \frac{1}{25}\left((5 \cdot 2 - 9)^5 - (5 \cdot 1 - 9)^5\right) = \frac{1}{25}(1^5 - (-4)^5) = \frac{1 - (-1024)}{25} = \frac{1025}{25} = 41$.
Ответ: $41$.

5) $\int_{-2,5}^{-2} \frac{8dx}{(2x + 3)^3} = 8\int_{-2,5}^{-2} (2x+3)^{-3} dx = 8\left(\frac{(2x+3)^{-2}}{-2 \cdot 2}\right) \Big|_{-2,5}^{-2} = -2\left[\frac{1}{(2x+3)^2}\right]_{-2,5}^{-2} = -2\left(\frac{1}{(2(-2)+3)^2} - \frac{1}{(2(-2,5)+3)^2}\right) = -2\left(\frac{1}{(-1)^2} - \frac{1}{(-2)^2}\right) = -2\left(1 - \frac{1}{4}\right) = -2\left(\frac{3}{4}\right) = -\frac{3}{2}$.
Ответ: $-\frac{3}{2}$.

6) $\int_{1}^{9} \left(1 + \frac{2}{\sqrt{x}}\right) dx = \int_{1}^{9} (1 + 2x^{-1/2}) dx = \left(x + \frac{2x^{1/2}}{1/2}\right) \Big|_{1}^{9} = \left(x + 4\sqrt{x}\right) \Big|_{1}^{9} = (9 + 4\sqrt{9}) - (1 + 4\sqrt{1}) = (9 + 12) - (1 + 4) = 21 - 5 = 16$.
Ответ: $16$.

7) $\int_{-1}^{1} \frac{dx}{\sqrt{4x + 5}} = \int_{-1}^{1} (4x+5)^{-1/2} dx = \left(\frac{(4x+5)^{1/2}}{1/2 \cdot 4}\right) \Big|_{-1}^{1} = \left(\frac{1}{2}\sqrt{4x+5}\right) \Big|_{-1}^{1} = \frac{1}{2}(\sqrt{4(1)+5} - \sqrt{4(-1)+5}) = \frac{1}{2}(\sqrt{9} - \sqrt{1}) = \frac{1}{2}(3-1) = 1$.
Ответ: $1$.

8) $\int_{0}^{6} \frac{dx}{\sqrt{4 - x/2}} = \int_{0}^{6} \left(4 - \frac{x}{2}\right)^{-1/2} dx = \left(\frac{(4 - x/2)^{1/2}}{1/2 \cdot (-1/2)}\right) \Big|_{0}^{6} = \left(-4\sqrt{4 - \frac{x}{2}}\right) \Big|_{0}^{6} = -4\left(\sqrt{4 - \frac{6}{2}} - \sqrt{4 - 0}\right) = -4(\sqrt{1} - \sqrt{4}) = -4(1-2) = 4$.
Ответ: $4$.

9) $\int_{\pi/6}^{\pi} \cos x dx = \left(\sin x\right) \Big|_{\pi/6}^{\pi} = \sin(\pi) - \sin\left(\frac{\pi}{6}\right) = 0 - \frac{1}{2} = -\frac{1}{2}$.
Ответ: $-\frac{1}{2}$.

10) $\int_{\pi/2}^{\pi} \sin \frac{x}{2} dx = \left(-\frac{\cos(x/2)}{1/2}\right) \Big|_{\pi/2}^{\pi} = \left(-2\cos\left(\frac{x}{2}\right)\right) \Big|_{\pi/2}^{\pi} = -2\left(\cos\left(\frac{\pi}{2}\right) - \cos\left(\frac{\pi}{4}\right)\right) = -2\left(0 - \frac{\sqrt{2}}{2}\right) = \sqrt{2}$.
Ответ: $\sqrt{2}$.

11) $\int_{3\pi/4}^{\pi} \frac{dx}{\cos^2 x} = (\tan x) \Big|_{3\pi/4}^{\pi} = \tan(\pi) - \tan\left(\frac{3\pi}{4}\right) = 0 - (-1) = 1$.
Ответ: $1$.

12) $\int_{\pi/12}^{\pi/9} \frac{dx}{\sin^2 3x} = \left(-\frac{\cot(3x)}{3}\right) \Big|_{\pi/12}^{\pi/9} = -\frac{1}{3}\left(\cot\left(\frac{3\pi}{9}\right) - \cot\left(\frac{3\pi}{12}\right)\right) = -\frac{1}{3}\left(\cot\left(\frac{\pi}{3}\right) - \cot\left(\frac{\pi}{4}\right)\right) = -\frac{1}{3}\left(\frac{1}{\sqrt{3}} - 1\right) = \frac{1}{3}\left(1 - \frac{\sqrt{3}}{3}\right) = \frac{3-\sqrt{3}}{9}$.
Ответ: $\frac{3-\sqrt{3}}{9}$.

13) $\int_{4}^{9} \sqrt{x} dx = \int_{4}^{9} x^{1/2} dx = \left(\frac{x^{3/2}}{3/2}\right) \Big|_{4}^{9} = \left(\frac{2}{3}x\sqrt{x}\right) \Big|_{4}^{9} = \frac{2}{3}(9\sqrt{9} - 4\sqrt{4}) = \frac{2}{3}(27 - 8) = \frac{38}{3}$.
Ответ: $\frac{38}{3}$.

14) $\int_{8}^{27} \sqrt[3]{x} dx = \int_{8}^{27} x^{1/3} dx = \left(\frac{x^{4/3}}{4/3}\right) \Big|_{8}^{27} = \left(\frac{3}{4}x\sqrt[3]{x}\right) \Big|_{8}^{27} = \frac{3}{4}(27\sqrt[3]{27} - 8\sqrt[3]{8}) = \frac{3}{4}(81 - 16) = \frac{3}{4}(65) = \frac{195}{4}$.
Ответ: $\frac{195}{4}$.

15) $\int_{-1}^{1} \sqrt{1-x} dx = \int_{-1}^{1} (1-x)^{1/2} dx = \left(-\frac{(1-x)^{3/2}}{3/2}\right) \Big|_{-1}^{1} = \left(-\frac{2}{3}(1-x)\sqrt{1-x}\right) \Big|_{-1}^{1} = -\frac{2}{3}(0 - (1-(-1))\sqrt{1-(-1)}) = -\frac{2}{3}(-2\sqrt{2}) = \frac{4\sqrt{2}}{3}$.
Ответ: $\frac{4\sqrt{2}}{3}$.

16) $\int_{\pi/3}^{2\pi/3} \cos\left(\frac{\pi}{3} - 3x\right) dx = \left(-\frac{1}{3}\sin\left(\frac{\pi}{3} - 3x\right)\right) \Big|_{\pi/3}^{2\pi/3} = -\frac{1}{3}\left(\sin\left(\frac{\pi}{3} - 2\pi\right) - \sin\left(\frac{\pi}{3} - \pi\right)\right) = -\frac{1}{3}\left(\sin\left(-\frac{5\pi}{3}\right) - \sin\left(-\frac{2\pi}{3}\right)\right) = -\frac{1}{3}\left(\frac{\sqrt{3}}{2} - \left(-\frac{\sqrt{3}}{2}\right)\right) = -\frac{1}{3}(\sqrt{3}) = -\frac{\sqrt{3}}{3}$.
Ответ: $-\frac{\sqrt{3}}{3}$.

17) $\int_{0}^{\ln 5} e^x dx = (e^x) \Big|_{0}^{\ln 5} = e^{\ln 5} - e^0 = 5 - 1 = 4$.
Ответ: $4$.

18) $\int_{2}^{3} 5^x dx = \left(\frac{5^x}{\ln 5}\right) \Big|_{2}^{3} = \frac{5^3 - 5^2}{\ln 5} = \frac{125 - 25}{\ln 5} = \frac{100}{\ln 5}$.
Ответ: $\frac{100}{\ln 5}$.

19) $\int_{-8}^{0} e^{-x/8} dx = \left(\frac{e^{-x/8}}{-1/8}\right) \Big|_{-8}^{0} = (-8e^{-x/8}) \Big|_{-8}^{0} = -8(e^0 - e^{-(-8)/8}) = -8(1 - e) = 8(e-1)$.
Ответ: $8(e-1)$.

20) $\int_{3}^{27} \frac{dx}{x} = (\ln|x|) \Big|_{3}^{27} = \ln 27 - \ln 3 = \ln\left(\frac{27}{3}\right) = \ln 9$.
Ответ: $\ln 9$.

21) $\int_{e}^{e^5} \frac{3}{x} dx = (3\ln|x|) \Big|_{e}^{e^5} = 3(\ln e^5 - \ln e) = 3(5 - 1) = 12$.
Ответ: $12$.

22) $\int_{-3}^{-1} \left(\frac{4}{x} - x\right) dx = \left(4\ln|x| - \frac{x^2}{2}\right) \Big|_{-3}^{-1} = \left(4\ln|-1| - \frac{(-1)^2}{2}\right) - \left(4\ln|-3| - \frac{(-3)^2}{2}\right) = \left(0 - \frac{1}{2}\right) - \left(4\ln 3 - \frac{9}{2}\right) = -\frac{1}{2} - 4\ln 3 + \frac{9}{2} = 4 - 4\ln 3$.
Ответ: $4 - 4\ln 3$.

23) $\int_{0}^{5} \frac{dx}{7x+5} = \left(\frac{1}{7}\ln|7x+5|\right) \Big|_{0}^{5} = \frac{1}{7}(\ln|7 \cdot 5 + 5| - \ln|7 \cdot 0 + 5|) = \frac{1}{7}(\ln 40 - \ln 5) = \frac{1}{7}\ln\left(\frac{40}{5}\right) = \frac{1}{7}\ln 8$.
Ответ: $\frac{1}{7}\ln 8$.

24) $\int_{-2}^{0} \frac{dx}{3x-2} = \left(\frac{1}{3}\ln|3x-2|\right) \Big|_{-2}^{0} = \frac{1}{3}(\ln|3 \cdot 0 - 2| - \ln|3(-2) - 2|) = \frac{1}{3}(\ln|-2| - \ln|-8|) = \frac{1}{3}(\ln 2 - \ln 8) = \frac{1}{3}\ln\left(\frac{2}{8}\right) = \frac{1}{3}\ln\left(\frac{1}{4}\right) = -\frac{1}{3}\ln 4$.
Ответ: $-\frac{1}{3}\ln 4$.