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) $$\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\frac{dx}{\sin^2 x}$$.

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

$$\int_{-\pi/4}^{\pi/4}\frac{dx}{\cos^2 x}=\tg x\Big|_{-\pi/4}^{\pi/4}=\tg\tfrac{\pi}{4}-\tg\left(-\tfrac{\pi}{4}\right)=2$$

$$\int_{-\pi/2}^{\pi/2}\cos x\,dx=\sin x\Big|_{-\pi/2}^{\pi/2}=\sin\tfrac{\pi}{2}-\sin\left(-\tfrac{\pi}{2}\right)=2$$

$$\int_{\pi/4}^{\pi/2}\frac{dx}{\sin^2 x}=-\ctg x\Big|_{\pi/4}^{\pi/2}=-\ctg\tfrac{\pi}{2}-\left(-\ctg\tfrac{\pi}{4}\right)=1$$

Ответ: а) $$1$$; б) $$2$$; в) $$2$$; г) $$1$$.