vault backup: 2023-10-03 08:10:20

AMiAL/ICT/Ćwiczenia/Zadania/Całki/Zadanie 5.md

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+# Obliczyć całki z funkcji wymiernych.
+## 1
+$$\int\frac{x+2}{x-3}dx=\int\frac{x-3}{x-3}+\frac{5}{x-3}dx=\int1+\frac{5}{x-3}dx=x+5\ln|x-3|+C$$
+## 2
+$$\int\frac{2x+3}{2x+5}dx=\int1+\frac{-2}{2x+5}dx=x-\log(x+\frac{5}{2})+C$$
+## 3 
+$$\int\frac{x^{2}dx}{x-1}=\int x^{2}\cdot\frac{1}{x-1}dx$$
+## 4
+$$\int\frac{x^{3}+x+1}{x-1}dx=\int\frac{x(x^2+1)+1}{x-1}dx=\int\frac{x(x^{2}+1)}{x-1}+\frac{1}{x-1}dx$$