Упр.11.19 ГДЗ Мерзляк 11 класс Углубленный уровень (Алгебра)

1) (0; п/12)tg^2 3xdx; 3) (0; п/2)cos(3x)cos(x)dx;
2) (-п; 0)2sin^2(x/4)dx; 4) (1; 2)(e^x+x^3)/(x^3 e^x)dx.

1. $$\int\limits_0^{\pi/12}\tg^2 3x\,dx=\int\limits_0^{\pi/12}\left(\frac{1}{\cos^2 3x}-1\right)dx$$
   $$=\frac13\tg 3x-x\Bigg|_0^{\pi/12}$$
   $$=\left(\frac13\tg\frac{\pi}{4}-\frac{\pi}{12}\right)-\left(\frac13\tg 0-0\right)=\frac13-\frac{\pi}{12}=\frac{4-\pi}{12}.$$

2. $$\int\limits_{-\pi}^{0}2\sin^2\frac{x}{4}\,dx=\int\limits_{-\pi}^{0}2\cdot\frac{1-\cos\frac{x}{2}}{2}\,dx$$
   $$=\int\limits_{-\pi}^{0}\left(1-\cos\frac{x}{2}\right)dx=x-2\sin\frac{x}{2}\Bigg|_{-\pi}^{0}$$
   $$=\left(0-2\sin 0\right)-\left(-\pi-2\sin\left(-\frac{\pi}{2}\right)\right)=\pi-2.$$

3. $$\int\limits_0^{\pi/2}\cos 3x\cos x\,dx=\int\limits_0^{\pi/2}\frac12(\cos 2x+\cos 4x)\,dx$$
   $$=\frac12\left(\frac12\sin 2x+\frac14\sin 4x\right)\Bigg|_0^{\pi/2}$$
   $$=\frac14\sin\pi+\frac18\sin 2\pi-\frac14\sin 0-\frac18\sin 0=0.$$

4. $$\int\limits_1^2\frac{e^x+x^3}{x^3e^x}\,dx=\int\limits_1^2\left(\frac1{x^3}+\frac1{e^x}\right)dx$$
   $$=\left(-\frac{1}{2x^2}-e^{-x}\right)\Bigg|_1^2$$
   $$=\left(-\frac18-\frac1{e^2}\right)-\left(-\frac12-\frac1e\right)=\frac38+\frac1e-\frac1{e^2}.$$
   $$\frac38+\frac1e-\frac1{e^2}=\frac{3e^2+8e-8}{8e^2}.$$

Ответ

1) $$\frac{4-\pi}{12}$$; 2) $$\pi-2$$; 3) $$0$$; 4) $$\frac{3e^2+8e-8}{8e^2}$$.