Упр.11.4 ГДЗ Мерзляк 11 класс Базовый уровень (Алгебра)

1) ?(-4; -2)2dx; 4) ?(?/4; ?/2)dx/sin^2(x); 7) ?(1; e)dx/x;
2) ?(1; 2)x^3dx; 5) ?(1; 3)dx/x^4; 8) ?(4; 9)vxdx;
3) ?(0; ?/2)cos(x)dx; 6) ?(0; 4)e^xdx; 9) ?(-1; 1)(1-5x^4)dx.

1. $$\int_{-4}^{-2} 2\,dx = 2x\Big|_{-4}^{-2} = 2\cdot(-2)-2\cdot(-4)= -4+8=4.$$
2. $$\int_{1}^{2} x^3\,dx = \frac{x^4}{4}\Big|_{1}^{2} = \frac{16}{4}-\frac{1}{4}=\frac{15}{4}.$$
3. $$\int_{0}^{\pi/2} \cos x\,dx = \sin x\Big|_{0}^{\pi/2} = \sin\frac{\pi}{2}-\sin 0=1.$$
4. $$\int_{\pi/4}^{\pi/2} \frac{dx}{\sin^2 x} = \int_{\pi/4}^{\pi/2} \csc^2 x\,dx = -\ctg x\Big|_{\pi/4}^{\pi/2} = -\ctg\frac{\pi}{2}+\ctg\frac{\pi}{4}=1.$$
5. $$\int_{1}^{3} \frac{dx}{x^4} = \int_{1}^{3} x^{-4}\,dx = \frac{x^{-3}}{-3}\Big|_{1}^{3} = -\frac{1}{3x^3}\Big|_{1}^{3} = -\frac{1}{3\cdot 27}+\frac{1}{3}=\frac{26}{81}.$$
6. $$\int_{0}^{4} e^x\,dx = e^x\Big|_{0}^{4} = e^4-e^0=e^4-1.$$
7. $$\int_{1}^{e} \frac{dx}{x} = \ln|x|\Big|_{1}^{e} = \ln e-\ln 1=1.$$
8. $$\int_{4}^{9} \sqrt{x}\,dx = \int_{4}^{9} x^{1/2}\,dx = \frac{2}{3}x^{3/2}\Big|_{4}^{9} = \frac{2}{3}\left(9^{3/2}-4^{3/2}\right)=\frac{2}{3}(27-8)=\frac{38}{3}.$$
9. $$\int_{-1}^{1} (1-5x^4)\,dx = \left(x-x^5\right)\Big|_{-1}^{1} = (1-1)-(-1-(-1))=0.$$

Ответ

1) $$4$$; 2) $$\frac{15}{4}$$; 3) $$1$$; 4) $$1$$; 5) $$\frac{26}{81}$$; 6) $$e^4-1$$; 7) $$1$$; 8) $$\frac{38}{3}$$; 9) $$0$$.