Integrali immediati

Una lista di integrali immediati

$$ \int x^n \ dx = \frac{x^{n+1}}{n+1} +c \ \ \ \ n \ne -1 $$

$$ \int \frac{1}{x} \ dx = \ln |x| + c $$

$$ \int a^x \ dx = \frac{ a^x}{\ln a} + c $$

$$ \int e^x \ dx = e^x + c $$

$$ \int \sin x \ dx = - \cos(x) + c $$

$$ \int \cos x \ dx = \sin(x) + c $$

$$ \int \frac{1}{\cos^2 x} \ dx = \tan(x) + c $$

$$ \int \frac{1}{\sin^2 x} \ dx = - \text{cotan}(x) + c $$

$$ \int \frac{1}{\sqrt{1-x^2}} \ dx = \arcsin(x) + c $$

$$ \int \frac{1}{\sqrt{a^2-x^2}} \ dx = \frac{ \arcsin(x) }{|a|} + c $$

$$ \int \frac{1}{1+x^2} \ dx = \arctan(x) + c $$

$$ \int \frac{1}{a^2+x^2} \ dx = \frac{1}{a} \cdot \arctan( \frac{x}{a} ) + c $$

Dai precedenti integrali immediati ottengo anche degli integrali generalizzati

$$ \int [f(x)]^n \ dx = \frac{[f(x)]^{n+1}}{n+1} +c \ \ \ \ n \ne -1 $$

$$ \int \frac{f'(x)}{f(x)} \ dx = \ln |f(x)| + c $$

$$ \int a^{f(x)} \cdot f'(x) \ dx = \frac{a^{f(x)}}{\ln a} + c $$

$$ \int e^{f(x)} \cdot f'(x) \ dx = e^{f(x)} + c $$

$$ \int \sin f(x) \cdot f'(x) \ dx = - \cos[f(x)] + c $$

$$ \int \cos f(x) \cdot f'(x) \ dx = \sin[f(x)] + c $$

$$ \int \frac{f'(x)}{\cos^2 f(x)} \ dx = \tan[f(x)] + c $$

$$ \int \frac{f'(x)}{\sin^2 f(x)} \ dx = - \text{cotan}[f(x)] + c $$

$$ \int \frac{f'(x)}{\sqrt{1-f(x)^2}} \ dx = \arcsin[f(x)] + c $$

$$ \int \frac{f'(x)}{\sqrt{a^2-f(x)^2}} \ dx = \frac{ \arcsin[f(x)] }{|a|} + c $$

$$ \int \frac{f'(x)}{1+f(x)^2} \ dx = \arctan[f(x)] + c $$

E così via