Question

Cum calculez integrala din x patrat +9 supra x ?

Answer 1

Faci impartirea, ca sa obtii 2 functii mai simple pt care cunosti in mod direct integrala:
$\frac{x^2+9}{x}= \frac{x^2}{x}+ \frac{9}{x} =x+ \frac{9}{x}$
∫(x + 9 / x)dx = ∫ x dx + 9·∫ (1 / x) dx
integrala e procesul invers derivarii, daca stii ca x² ' = 2x, atunci ∫2x dx = x²
(x² / 2)' = 2x / 2 = x; <=> ∫ x dx = x² / 2 + C
(ln x)' = 1 / x; <=> ∫(1 / x) dx = ln(|x|) + C
deci ∫ x dx + 9·∫ (1 / x) dx = x² / 2 + 9·ln(|x|) + C
in caz ca nu te-ai referit la $\frac{x^2+9}{x}$ ci la x² + 9 / x, atunci raspunsul e:
∫(x² + 9/x)dx = ∫x² dx + 9·∫(1/x)dx = x³ / 3 + 9·ln(|x|) + C

Answer 2

$\it \int(x^2+\dfrac{9}{x}) dx = \int x^2 dx +\int \dfrac{9}{x} dx = \dfrac{x^3}{3} + 9 ln|x| + \mathcal{C}$