Упр.1005 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\int_{1}^{e}\frac{dx}{x}$$
2. $$\int_{0}^{\ln 2}e^x\,dx$$
3. $$\int_{-\pi}^{2\pi}\cos x\,dx$$
4. $$\int_{-2\pi}^{\pi}\sin x\,dx$$
5. $$\int_{-2\pi}^{\pi}\sin 2x\,dx$$
6. $$\int_{-3\pi}^{0}\cos 3x\,dx$$

1. $$\int\limits_{1}^{e}\frac{1}{x}\,dx=\ln x\Big|_{1}^{e}=\ln e-\ln 1=1-0=1.$$

2. $$\int\limits_{0}^{\ln 2}e^x\,dx=e^x\Big|_{0}^{\ln 2}=e^{\ln 2}-e^0=2-1=1.$$

3. $$\int\limits_{-\pi}^{2\pi}\cos x\,dx=\sin x\Big|_{-\pi}^{2\pi}=\sin 2\pi-\sin(-\pi)=0-0=0.$$

4. $$\int\limits_{-2\pi}^{\pi}\sin x\,dx=-\cos x\Big|_{-2\pi}^{\pi}=-\cos\pi+\cos(-2\pi)=1+1=2.$$

5. $$\int\limits_{-2\pi}^{\pi}\sin 2x\,dx=-\frac{1}{2}\cos 2x\Big|_{-2\pi}^{\pi}=-\frac{1}{2}\bigl(\cos 2\pi-\cos(-4\pi)\bigr)=-\frac{1}{2}(1-1)=0.$$

6. $$\int\limits_{-3\pi}^{0}\cos 3x\,dx=\frac{1}{3}\sin 3x\Big|_{-3\pi}^{0}=\frac{1}{3}\bigl(\sin 0-\sin(-9\pi)\bigr)=0.$$

Ответ: 1) $$1$$; 2) $$1$$; 3) $$0$$; 4) $$2$$; 5) $$0$$; 6) $$0$$.