OBLICZANIE PROSTYCH CAŁEK - ZADANIA
Zadanie 9
Oblicz całkę \(\displaystyle\int \frac{2x^2-3\sqrt[3]x}{x}\,dx\)
Rozwiązanie
\(\displaystyle\int \frac{2x^2-3\sqrt[3]x}{x}\,dx=\)
\(\displaystyle\int \frac{2x^2}{x}-\frac {3\sqrt[3]x}{x}\,dx=\)
\(\displaystyle\int \left(f(x)+g(x)\right)dx=\int f(x)dx+\int g(x)dx\)
\(\displaystyle 2\int \frac{x^2}{x}dx-3\int\frac {\sqrt[3]x}{x}\,dx=\)
\(\sqrt[a]x=a^{\frac1a}\) \(\frac{x^n}{x^m}=x^{n-m}\)
\(\displaystyle 2\int x^{2-1}\,dx- 3\int x^{\frac13-1}\,dx=\)
\(\displaystyle 2\int x^1\,dx- 3\int x^{-\frac23}\,dx=\)
\( \int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c\)
\(\displaystyle 2\frac{x^{1+1}}{1+1}-3\frac{x^{-\frac23+1}}{-\frac23+1}+c=\)
\(\displaystyle 2\frac{x^{2}}{2}-3\frac{x^{\frac13}}{\frac13}+c=\)
\(\displaystyle x^{2}-9x^{\frac13}+c=\)
\(\displaystyle x^{2}-9\sqrt[3]x+c\)
Odpowiedź
\(\displaystyle\int \frac{2x^2-3\sqrt[3]x}{x}\,dx=x^{2}-9\sqrt[3]x+c\)
