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

1) ?(5; 7)xdx; 6) ?(?/4; ?/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; ?/2)(4sin(x)+2cos(x))dx.
5) ?(0; ?/3)sin(x)dx; 10) ?(-2; 3)3^xdx;

1. $$\int\limits_{5}^{7} x\,dx=\left.\frac{x^2}{2}\right|_{5}^{7}=\frac{49}{2}-\frac{25}{2}=12.$$
2. $$\int\limits_{3}^{8} 2\,dx=\left.2x\right|_{3}^{8}=16-6=10.$$
3. $$\int\limits_{-3}^{0} x^2\,dx=\left.\frac{x^3}{3}\right|_{-3}^{0}=0-\left(-\frac{27}{3}\right)=9.$$
4. $$\int\limits_{-1}^{2} x^4\,dx=\left.\frac{x^5}{5}\right|_{-1}^{2}=\frac{32}{5}-\left(-\frac{1}{5}\right)=\frac{33}{5}.$$
5. $$\int\limits_{0}^{\pi/3} \sin x\,dx=\left.-\cos x\right|_{0}^{\pi/3}=-\frac12-(-1)=\frac12.$$
6. $$\int\limits_{\pi/4}^{\pi/3} \frac{dx}{\cos^2 x}=\left.\tan x\right|_{\pi/4}^{\pi/3}=\sqrt{3}-1.$$
7. $$\int\limits_{16}^{100} \frac{dx}{\sqrt{x}}=\int\limits_{16}^{100} x^{-1/2}\,dx=\left.2\sqrt{x}\right|_{16}^{100}=20-8=12.$$
8. $$\int\limits_{e^2}^{e^3} \frac{dx}{x}=\left.\ln|x|\right|_{e^2}^{e^3}=3-2=1.$$
9. $$\int\limits_{1}^{10} \frac{dx}{x^2}=\int\limits_{1}^{10} x^{-2}\,dx=\left.-\frac{1}{x}\right|_{1}^{10}=-\frac{1}{10}+1=\frac{9}{10}.$$
10. $$\int\limits_{-2}^{3} 3^x\,dx=\left.\frac{3^x}{\ln 3}\right|_{-2}^{3}=\frac{27}{\ln 3}-\frac{1}{9\ln 3}=\frac{242}{9\ln 3}.$$
11. $$\int\limits_{1}^{8} \sqrt[3]{x}\,dx=\int\limits_{1}^{8} x^{1/3}\,dx=\left.\frac{3}{4}x^{4/3}\right|_{1}^{8}=\frac{3}{4}\cdot 16-\frac{3}{4}=\frac{45}{4}.$$
12. $$\int\limits_{-4}^{-2} (2x+4)\,dx=\left.(x^2+4x)\right|_{-4}^{-2}=(-4)-0=-4.$$
13. $$\int\limits_{0}^{6} (3x^2-x)\,dx=\left.\left(x^3-\frac{x^2}{2}\right)\right|_{0}^{6}=216-18=198.$$
14. $$\int\limits_{0}^{\pi/2} (4\sin x+2\cos x)\,dx=\left.(-4\cos x+2\sin x)\right|_{0}^{\pi/2}=6.$$

Ответ

1) $$12$$; 2) $$10$$; 3) $$9$$; 4) $$\frac{33}{5}$$; 5) $$\frac12$$; 6) $$\sqrt3-1$$; 7) $$12$$; 8) $$1$$; 9) $$\frac{9}{10}$$; 10) $$\frac{242}{9\ln 3}$$; 11) $$\frac{45}{4}$$; 12) $$-4$$; 13) $$198$$; 14) $$6$$.