Упр.11.3 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

1) (5; 7)xdx; 6) (п/4; п/3)dx/cos^2(x); 1) (1; 8)x^(1/3)dx;
2) (3; 8)dx; 7) (16; 100)dx/vx; 12) (-4; -2)(2x+4)dx;
3) (-3; 0)x^2dx; 8) (e^2; e^3)dx/x; 13) (0; 6)(3x^2-x)dx;
4) (-1; 2)x^4dx; 9) (1; 10)dx/x^2; 14) (0; п/2)(4sin(x)+2cos(x))dx.
5) (0; п/3)sin(x)dx; 10) (-2; 3)3^xdx;

1. $$\int_{5}^{7} x\,dx=\left.\frac{x^2}{2}\right|_{5}^{7}=\frac{49}{2}-\frac{25}{2}=12.$$
2. $$\int_{3}^{8} dx=\left.x\right|_{3}^{8}=8-3=5.$$
3. $$\int_{-3}^{0} x^2\,dx=\left.\frac{x^3}{3}\right|_{-3}^{0}=0-\left(-\frac{27}{3}\right)=9.$$
4. $$\int_{-1}^{2} x^4\,dx=\left.\frac{x^5}{5}\right|_{-1}^{2}=\frac{32}{5}-\left(-\frac{1}{5}\right)=\frac{33}{5}.$$
5. $$\int_{0}^{\pi/3} \sin x\,dx=\left.-\cos x\right|_{0}^{\pi/3}=-\cos\frac{\pi}{3}+\cos 0=\frac{1}{2}.$$
6. $$\int_{\pi/4}^{\pi/3}\frac{dx}{\cos^2 x}=\left.\tan x\right|_{\pi/4}^{\pi/3}=\tan\frac{\pi}{3}-\tan\frac{\pi}{4}=\sqrt{3}-1.$$
7. $$\int_{16}^{100}\frac{dx}{\sqrt{x}}=\int_{16}^{100}x^{-1/2}\,dx=\left.2\sqrt{x}\right|_{16}^{100}=2\cdot 10-2\cdot 4=12.$$
8. $$\int_{e^2}^{e^3}\frac{dx}{x}=\left.\ln|x|\right|_{e^2}^{e^3}=3-2=1.$$
9. $$\int_{1}^{10}\frac{dx}{x^2}=\int_{1}^{10}x^{-2}\,dx=\left.-\frac{1}{x}\right|_{1}^{10}=-\frac{1}{10}+1=\frac{9}{10}.$$
10. $$\int_{-2}^{3}3^x\,dx=\left.\frac{3^x}{\ln 3}\right|_{-2}^{3}=\frac{27}{\ln 3}-\frac{1}{9\ln 3}=\frac{242}{9\ln 3}.$$
11. $$\int_{1}^{8}\sqrt[3]{x}\,dx=\int_{1}^{8}x^{1/3}\,dx=\left.\frac{3}{4}x^{4/3}\right|_{1}^{8}=\frac{3}{4}\cdot 16-\frac{3}{4}=\frac{45}{4}.$$
12. $$\int_{-4}^{-2}(2x+4)\,dx=\left.(x^2+4x)\right|_{-4}^{-2}=(-4)-0=-4.$$
13. $$\int_{0}^{6}(3x^2-x)\,dx=\left.(x^3-\frac{x^2}{2})\right|_{0}^{6}=216-\frac{36}{2}=198.$$
14. $$\int_{0}^{\pi/2}(4\sin x+2\cos x)\,dx=\left.(-4\cos x+2\sin x)\right|_{0}^{\pi/2}=(-4\cdot 0+2\cdot 1)-(-4\cdot 1+2\cdot 0)=6.$$

Ответ

1) 12; 2) 5; 3) 9; 4) $$\frac{33}{5}$$; 5) $$\frac{1}{2}$$; 6) $$\sqrt{3}-1$$; 7) 12; 8) 1; 9) $$\frac{9}{10}$$; 10) $$\frac{242}{9\ln 3}$$; 11) $$\frac{45}{4}$$; 12) -4; 13) 198; 14) 6.