Упр.11.20 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

Задача

1) (5п/4; 15п/4)ctg^2(x/5)dx; 3) (-п/2; -п/4)sin(7x)cos(3x)dx;
2) (-п/2; п/2)cos^2(2x)dx; 4) (-2; -1)(x^2-e^x)/(x^2 e^x)dx.

Подробный ответ

$$\int\limits_{5\pi/4}^{15\pi/4}\ctg^2\frac{x}{5}\,dx=\int\limits_{5\pi/4}^{15\pi/4}\left(\frac{1}{\sin^2\frac{x}{5}}-1\right)\,dx$$
$$=-5\ctg\frac{x}{5}-x\Bigg|_{5\pi/4}^{15\pi/4}$$
$$=\left(-5\ctg\frac{3\pi}{4}-\frac{15\pi}{4}\right)-\left(-5\ctg\frac{\pi}{4}-\frac{5\pi}{4}\right)$$
$$=10-\frac{10\pi}{4}=10-\frac{5\pi}{2}=\frac{20-5\pi}{2}.$$
$$\int\limits_{-\pi/2}^{\pi/2}\cos^2 2x\,dx=\int\limits_{-\pi/2}^{\pi/2}\frac{1+\cos 4x}{2}\,dx$$
$$=\frac12\left(x+\frac14\sin 4x\right)\Bigg|_{-\pi/2}^{\pi/2}$$
$$=\frac12\left(\frac{\pi}{2}+\frac14\sin 2\pi\right)-\frac12\left(-\frac{\pi}{2}+\frac14\sin(-2\pi)\right)=\frac{\pi}{2}.$$
$$\int\limits_{-\pi/2}^{-\pi/4}\sin 7x\cos 3x\,dx=\int\limits_{-\pi/2}^{-\pi/4}\frac12(\sin 4x+\sin 10x)\,dx$$
$$=\frac12\left(-\frac14\cos 4x-\frac1{10}\cos 10x\right)\Bigg|_{-\pi/2}^{-\pi/4}$$
$$=\left(-\frac18\cos\pi-\frac1{20}\cos\frac{5\pi}{2}\right)-\left(-\frac18\cos 2\pi-\frac1{20}\cos 5\pi\right)$$
$$=\frac18-\left(-\frac18+\frac1{20}\right)=\frac15.$$
$$\int\limits_{-2}^{-1}\frac{x^2-e^x}{x^2e^x}\,dx=\int\limits_{-2}^{-1}\left(\frac1{e^x}-\frac1{x^2}\right)\,dx$$
$$=\int\limits_{-2}^{-1}\left(e^{-x}-x^{-2}\right)\,dx=-e^{-x}+\frac1x\Bigg|_{-2}^{-1}$$
$$=\left(-e^1-1\right)-\left(-e^2-\frac12\right)=e^2-e-\frac12.$$