Tabella degli integrali
∫ f x integrale F x primitiva ∫ f x integrale F x primitiva
∫x dx x22 c ∫ ±1
1−x2dx {±arcsin x c
∓arccos x c
∫a dx ax c ∫ 1
x2−1dx log∣xx2−1∣ c
∫axdx log aax c ∫ 1
axdx − a−x
log a c
∫ x
x21dx 1
2log∣x21∣ c ∫ 1
xndx −n−1
xn−1 c
∫a⋅xndx a⋅xn1n1 c ∫ 1
ax2dx a0 1
aarctan
x
a c
∫1xdx log∣x∣c ∫ 1
1−x2dx 1
2log∣1 x1− x∣ c
∫1xdx 2x c ∫ 1
1x2dx {arcSh x c
logx1x2 c
∫sin x dx −cos x c ∫sin2x dx 12x−sin x cos x c
∫cos x dx sin x c ∫cos2x dx 12xsin x cos x c
∫tan x dx −logcos x c ∫tan x1 dx logsin x c
∫arcsin x dx 1−x2x arcsin x c ∫ 1
x2±a2 dx log∣xx±a2∣ c
∫arccos x dx x arccos x−1− x2 c ∫x2±a2dx 2xx2±a2±a2
2logxx2±a2 c
∫e±k xdx ±e±k xk c ∫ 1
ek xdx −e−k x
k c
∫1tan2xdx =
∫ 1
cos2x dx tan x c ∫cos x1 dx log∣tanx2
4∣ c
∫1ctg2xdx =
∫ 1
sin2xdx −ctg x c ∫sin x1 dx log∣tan2x∣ c
∫Sh x dx Ch x c ∫a2−x2dx 12a2arcsinxaxa2−x2 c
∫Ch x dx Sh xc ∫ 1
Ch2x dx =
∫1−Th2xdx c
Th x c
∫ 2x
x21dx log x21 c ∫ 1
x2a2dx 1
aarctanx a c
Proprietà
∫k⋅ f x dx = k⋅∫ f x dx
∫ f x g x ... fnx dx = ∫ f x dx ∫g x dx ... fnx dx
∫ f x dx = a∫1a f x dx = 1
a∫a f x dx = ∫aa f x dx a∈R
Integrali indefiniti riconducibili ad elementari
∫ f x integrale F x primitiva
∫ fnx⋅ f ' x dx f n1x
n1 c
∫ f ' xf x dx log∣ f x∣ c
∫ f ' x ⋅cos f x dx sin f x c
∫ f ' x ⋅sin f x dx −cos f x c
∫ef x f ' x dx ef x c
∫af x f ' x dx af x
ln a c
∫ f ' x
1− f2xdx { arcsin f x c
−arccos f x c
∫ f ' x
1 f2xdx arctan f x c
Integrale definito
∫
a b
f x dx = F b − F a = [F x ]ab
dove F è la primitiva di f(x)
Integrazione per parti Integrale
indefinito ∫ f x g ' x dx = f x g x −∫f ' x g x dx
f(x) va derivata e g'(x) va integrata
Integrale
definito ∫
a b
f x g ' x dx = [ f b ⋅ g b − f a ⋅ g a ] − ∫
a b
f ' x g x dx
Si integrano per parti funzioni del tipo:
P x ⋅ex P x ⋅sin x P x cos x ex⋅sin x ex⋅cos x
dove P(x) è un polinomio
Integrazione per sostituzione
Integrale indefinito ∫ f h x h ' x dx = ∫ f y dyy=h x
Integrale
definito ∫
a b
f h x h ' x dx = ∫
h a h b
f y dy