Sei sulla pagina 1di 4

Integrali

Integrale Integrale
Primitive Primitive
(forma semplice) (forma generale)

∫ 0 dx 0

∫ k dx,  k ∈ R kx + c
α+1 α+1
α x α ′ [f (x)]
∫ x  dx, α ∈ R ∖ {−1} + c ∫ [f (x)] ⋅ f (x) dx, α ∈ R ∖ {−1}
α+1 α+1

1 1 ′
∫  dx ∫ ⋅ f (x) dx
√x + c √f (x) + c
2 √x 2√f (x)

x x f (x) ′ f (x)
∫ e  dx e + c ∫ e ⋅ f (x) dx e + c

x f (x)
x a f (x) ′ a
∫ a  dx + c ∫ a ⋅ f (x) dx + c
ln a ln a

1 1 ′
∫  dx ln |x| + c ∫ ⋅ f (x) dx ln |f (x)| + c
x f (x)

∫ cos x dx sin x + c ∫ cos[f (x)] ⋅ f (x) dx sin f (x) + c

∫ sin x dx − cos x + c ∫ sin[f (x)] ⋅ f (x) dx − cos f (x) + c

1 f (x)
∫  dx tan x + c ∫  dx tan f (x) + c
2 2
cos x cos f (x)

1 f (x)
∫ −  dx cot x + c ∫ −  dx cot f (x) + c
2 2
sin x sin f (x)

1 1 ′
∫  dx arcsin x + c ∫ ⋅ f (x) dx arcsin f (x) + c
√1−x2 √1−[f (x)]2

1 1 ′
∫ −  dx arccos x + c ∫ − ⋅ f (x) dx arccos f (x) + c
√1−x2 √1−[f (x)]2

1 1 ′
∫  dx arctan x + c ∫ ⋅ f (x) dx arctan f (x) + c
2 2
1+x 1+[f (x)]

1 1 ′
∫ −  dx arccot x + c ∫ − ⋅ f (x) dx arccot f (x) + c
2 2
1+x 1+[f (x)]

∫ cosh x dx sinh x + c ∫ cosh[f (x)] ⋅ f (x) dx sinh f (x) + c

∫ sinh x dx cosh x + c ∫ sinh[f (x)] ⋅ f (x) dx cosh f (x) + c

1 f (x)
∫  dx tanh x + c ∫  dx tanh f (x) + c
2 2
cosh x cosh f (x)

1 f (x)
∫  dx coth x + c ∫  dx coth f (x) + c
2 2
sinh x sinh f (x)

Teoremi sul calcolo integrale

INTEGRALE DEL PRODOTTO DI UNA COSTANTE k PER UNA FUNZIONE f :

∫ k ⋅ f (x) dx = k ⋅ ∫ f (x) dx

22
www.webtutordimatematica.it
INTEGRALE DELLA SOMMA DI DUE FUNZIONI f  e g:

∫ f (x) + g(x) dx = ∫ f (x) dx + ∫ g(x) dx

METODO DI INTEGRAZIONE PER PARTI:

′ ′
∫ f (x) ⋅ g(x) dx = f (x) ⋅ g(x) − ∫ f (x) ⋅ g (x) dx

ESTENSIONE DEL CONCETTO DI INTEGRALE:
b a

∫ f (x) dx = − ∫ f (x) dx
a b

PROPRIETA' ADDITIVA DELL'INTEGRALE (c ∈ [a, b] ):

b c b

∫ f (x) dx = ∫ f (x) dx + ∫ f (x) dx

a a c

TEOREMA FONDAMENTALE DEL CALCOLO INTEGRALE:

Se f (x) : [a, b] → R  è continua in [a, b], allora, per ogni x ∈ [a, b]  si ha:

x

F (x) = ∫ f (x) dx ⇒ F (x) = f (x)
a

TEOREMA DELLA MEDIA:

Se f (x) : [a, b] → R  è continua in [a, b], esiste c ∈ [a, b]  tale che:

∫ f (x) dx = (b − a) ⋅ f (c)
a

Integrali impropri

b b

+
a x0 →a x0

Se f (x) : (a, b) → R  è continua in [a, b[, si ha:

23
www.webtutordimatematica.it
b x0

x0 →b
a a

b x0 b

x0 →c +
a x0 →c x0
a

+∞ x0

x0 →+∞
a a

b b

x0 →−∞
−∞ x0

+∞ x2

∫ f (x) dx = lim ∫ f (x) dx

−∞ x1 →−∞ x1

x2 →+∞

Criteri di integrabilità
Se f (x) è una funzione continua in [a, b[ e se

b
⎧0 con p < 1,  allora  ∫ f (x) dx converge

⎪ a
p b
lim (b − x) f (x) = ⎨ con p ≥ 1,  allora  ∫ f (x) dx diverge

x→b a

⎪ b
l ∈ R ∖ {0}  allora  ∫ f (x) dx converge se e solo se p < 1
a

Se f (x) è una funzione continua in ]a, b] e se

b
⎧0 con p < 1,  allora  ∫ f (x) dx converge

⎪ a
p b
lim (x − a) f (x) = ⎨ con p ≥ 1,  allora  ∫ f (x) dx diverge

+ a
x→a

⎪ b
l ∈ R ∖ {0}  allora  ∫ f (x) dx converge se e solo se p < 1
a

Se f (x) è una funzione continua in [a, +∞[ e se

+∞

⎪0 con p > 1,  allora  ∫ f (x) dx converge
a
p +∞
lim x f (x) = ⎨ ∞ con p ≤ 1,  allora  ∫ f (x) dx diverge
x→+∞ a

⎩ +∞
l ∈ R ∖ {0}  allora  ∫ f (x) dx converge se e solo se p > 1
a

24
www.webtutordimatematica.it
Funzione inversa e retta tangente

Una funzione strettamente monotona (crescente o decrescente) è invertibile.

Se una funzione f (x)  è invertibile e derivabile in x 0  con y0 = f (x 0 ) , allora la sua

derivata prima nel punto sarà:

1
−1
Df (y 0 ) =

f (x 0 )

L'equazione della retta tangente al grafico della funzione y = f (x)  nel punto x 0  è

y − f (x 0 ) = f (x 0 )(x − x 0 )

dove f ′ (x 0 ) è la derivata della funzione f  calcolata nel punto x = x0 .

25
www.webtutordimatematica.it
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://it.scribd.com/document/551127931/formulario-integrali", "fetch_time": 1695552533000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2023-40/segments/1695233506632.31/warc/CC-MAIN-20230924091344-20230924121344-00063.warc.gz", "warc_record_offset": 360227952, "warc_record_length": 208859, "token_count": 2158, "char_count": 3651, "score": 3.609375, "int_score": 4, "crawl": "CC-MAIN-2023-40", "snapshot_type": "latest", "language": "en", "language_score": 0.22245937585830688, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00044-of-00064.parquet"}