Упр.49.2 ГДЗ Мордкович 10-11 класс (Алгебра)

а) $$\int_{\frac{\pi}{2}}^{\pi}\sin x\,dx$$
б) $$\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{dx}{\cos^2 x}$$
в) $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos x\,dx$$
г) $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{dx}{\sin^2 x}$$

Воспользуемся формулой Ньютона–Лейбница:

$$\int_a^b f(x)\,dx = F(x)\bigg|_a^b = F(b)-F(a).$$

а)

$$\int_{\pi/2}^{\pi}\sin x\,dx = -\cos x\bigg|_{\pi/2}^{\pi} = -\cos\pi — \bigl(-\cos\tfrac{\pi}{2}\bigr) = -(-1)-0=1.$$

Ответ: $$1$$

б)

$$\int_{-\pi/4}^{\pi/4}\frac{dx}{\cos^2 x} = \tg x\bigg|_{-\pi/4}^{\pi/4} = \tg\tfrac{\pi}{4}-\tg\bigl(-\tfrac{\pi}{4}\bigr)=1-(-1)=2.$$

Ответ: $$2$$

в)

$$\int_{-\pi/2}^{\pi/2}\cos x\,dx = \sin x\bigg|_{-\pi/2}^{\pi/2} = \sin\tfrac{\pi}{2}-\sin\bigl(-\tfrac{\pi}{2}\bigr)=1-(-1)=2.$$

Ответ: $$2$$

г)

$$\int_{\pi/4}^{\pi/2}\frac{dx}{\sin^2 x} = -\ctg x\bigg|_{\pi/4}^{\pi/2} = -\ctg\tfrac{\pi}{2}-\bigl(-\ctg\tfrac{\pi}{4}\bigr)=0-(-1)=1.$$

Ответ: $$1$$