Question

Тригонометрическая и показательны форма , и номер 6

Answer 1

4

$z = 3 - 3i$

$|z| = \sqrt{3^2 + (-3)^2} = 3\sqrt{2} \\\begin{cases} \cos \arg z = \frac{3}{3\sqrt{2}} = \frac{1}{\sqrt{2}} \\ \sin \arg z = \frac{-3}{3\sqrt{2}} = -\frac{1}{\sqrt{2}}\end{cases} \Rightarrow \arg z = -\frac{\pi}{4}$

$z = 3\sqrt{2} (\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))$

$e^{i\phi} = \cos(\phi) + i\sin(\phi) \Rightarrow z = 3\sqrt{2}\,e^{-\frac{i\pi}{4}}$

6

$\Large \displaystyle {\int {\frac{dx}{(5x + 13)^3}}} = \Large \displaystyle {\frac{1}{5}\int {\frac{d(5x + 13)}{(5x + 13)^3}}} = \begin{vmatrix}t = 5x + 13 \\ dt = 5\,dx\end{vmatrix} = \Large \displaystyle {\frac{1}{5}\int {\frac{dt}{t^3}}} =\\= -\frac{1}{10t^2} + C = -\frac{1}{10(5x + 13)^2} + C$