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

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

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

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

3. $$\int\limits_0^{\pi/2}\cos 3x\cos x\,dx=\int\limits_0^{\pi/2}\frac{1}{2}\left(\cos 2x+\cos 4x\right)dx$$
   $$=\frac{1}{2}\left(\frac{1}{2}\sin 2x+\frac{1}{4}\sin 4x\right)\Bigg|_0^{\pi/2}$$
   $$=\frac{1}{4}\sin\pi+\frac{1}{8}\sin 2\pi-\frac{1}{4}\sin 0-\frac{1}{8}\sin 0=0.$$

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

Ответ

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