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assets/notes/matematyka/matematyka_2023.12.18.md

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+##### Całkowanie ułamków prostych
+
+$$
+\frac{Ax+B}{((x-p)^2 +q)^k}
+$$
+
+```{tip}
+$$
+\frac{x+3}{((x-2)^2+5)^2} \\
+y=(x-2)^2+5 \\
+dy = 2(x-2)dx \\
+\frac{2(x-2)-2(x-2) + x+3}{((x-2)^2+5)^2} \\
+$$
+```
+
+```{note}
+Rozkładanie na ułamki proste
+
+$$
+\frac{1}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2} \\
+1 = A(x+2) + B(x-1) \\
+1 = Ax + 2A + Bx - B \\
+\left\{\begin{matrix}
+A + B = 0 \\
+2A - B = 1 \\
+\end{matrix}\right. \Rightarrow A = -B = \frac{1}{3} \\
+$$
+```
+
+#### Całkowanie Funkcji typu $R(cos(x), sin(x))$
+
+Metoda: Podstawienie $t=tg\frac{x}{2}$
+
+$$
+sin(x) = 2 * sin(\frac{x}{2}})cos(\frac{x}{2}) = \frac{2tg\frac{x}{2}}{1+tg^2\frac{x}{2}} \\
+cos(x) = \frac{1-t^2}{1+t^2} \\
+tg(x) = \frac{2t}{1+t} \\
+dt = 2 \frac{1}{1+t^2}dt
+$$
+
+#### Całki z pierwiastkiem
+
+$$
+\int \sqrt{1-x^2}dx \\
+x = sin(t) \\
+dx = cos(t) \\
+\int \sqrt{1-sin^2(t)} cos(t) dt \\
+\int cos^2(t) dt \\
+\\
+\\
+\int \sqrt{1+x^2}dx \\
+x= sinh(t) \\
+
+$$