Tabella di Integrali Definiti Importanti

1. Integrali con funzioni esponenziali

1.1 Integrali della forma xne−axx^n e^{-ax}

∫0∞xne−ax dx=Γ(n+1)an+1,a>0, n>−1.\int_0^\infty x^n e^{-ax} \, dx = \frac{\Gamma(n+1)}{a^{n+1}}, \quad a > 0,\ n > -1.

Per nn intero positivo:

∫0∞xne−ax dx=n!an+1.\int_0^\infty x^n e^{-ax} \, dx = \frac{n!}{a^{n+1}}.

∫0∞xne−ax2 dx=Γ(n+12)2a(n+1)/2,a>0, n>−1.\int_0^\infty x^n e^{-ax^2} \, dx = \frac{\Gamma\left(\frac{n+1}{2}\right)}{2a^{(n+1)/2}}, \quad a > 0,\ n > -1.

Casi particolari:

  • Se n=2kn = 2k (pari):

    ∫0∞x2ke−ax2 dx=(2k−1)!!π2k+1ak+1/2.\int_0^\infty x^{2k} e^{-ax^2} \, dx = \frac{(2k-1)!! \sqrt{\pi}}{2^{k+1} a^{k+1/2}}.

  • Se n=2k+1n = 2k+1 (dispari):

    ∫0∞x2k+1e−ax2 dx=k!2ak+1.\int_0^\infty x^{2k+1} e^{-ax^2} \, dx = \frac{k!}{2a^{k+1}}.

1.2 Integrali gaussiani

∫0∞e−a2x2 dx=π2a,a>0.\int_0^\infty e^{-a^2 x^2} \, dx = \frac{\sqrt{\pi}}{2a}, \quad a > 0.

∫0∞x2e−a2x2 dx=π4a3,a>0.\int_0^\infty x^2 e^{-a^2 x^2} \, dx = \frac{\sqrt{\pi}}{4a^3}, \quad a > 0.

∫0∞e−a2x2cos⁡(bx) dx=π2ae−b2/(4a2),a>0.\int_0^\infty e^{-a^2 x^2} \cos(bx) \, dx = \frac{\sqrt{\pi}}{2a} e^{-b^2/(4a^2)}, \quad a > 0.

1.3 Integrali con denominatori esponenziali

∫0∞xex−1 dx=π26.\int_0^\infty \frac{x}{e^x - 1} \, dx = \frac{\pi^2}{6}.

∫0∞xex+1 dx=π212.\int_0^\infty \frac{x}{e^x + 1} \, dx = \frac{\pi^2}{12}.

∫0∞e−axsin⁡xx dx=arctan⁡(1a),a>0.\int_0^\infty \frac{e^{-ax} \sin x}{x} \, dx = \arctan\left(\frac{1}{a}\right), \quad a > 0.

1.4 Integrali con logaritmi

∫0∞e−xln⁡x dx=−γ,\int_0^\infty e^{-x} \ln x \, dx = -\gamma,

dove γ≈0,5772\gamma \approx 0,5772 è la costante di Eulero-Mascheroni.

2. Integrali con funzioni trigonometriche

2.1 Integrali di potenze di seno e coseno

∫0π/2sin⁡2α+1xcos⁡2β+1x dx=Γ(α+1)Γ(β+1)2Γ(α+β+2)=12B(α+1,β+1),\int_0^{\pi/2} \sin^{2\alpha+1} x \cos^{2\beta+1} x \, dx = \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{2\Gamma(\alpha+\beta+2)} = \frac{1}{2} B(\alpha+1, \beta+1),

valida per α>−1,β>−1\alpha > -1, \beta > -1.

2.2 Integrali della forma sin⁡(ax)x\frac{\sin(ax)}{x}

∫0∞sin⁡(ax)x dx={π2,a>0,−π2,a<0.\int_0^\infty \frac{\sin(ax)}{x} \, dx = \begin{cases} \frac{\pi}{2}, & a > 0, \\ -\frac{\pi}{2}, & a < 0. \end{cases}

∫0∞tan⁡(ax)x dx={π2,a>0,−π2,a<0.\int_0^\infty \frac{\tan(ax)}{x} \, dx = \begin{cases} \frac{\pi}{2}, & a > 0, \\ -\frac{\pi}{2}, & a < 0. \end{cases}

2.3 Integrali con combinazioni trigonometriche

∫0∞cos⁡(ax)−cos⁡(bx)x dx=ln⁡(ba),a,b>0.\int_0^\infty \frac{\cos(ax) - \cos(bx)}{x} \, dx = \ln\left(\frac{b}{a}\right), \quad a,b > 0.

∫0∞sin⁡xcos⁡(ax)x dx={π2,∣a∣<1,π4,∣a∣=1,0,∣a∣>1.\int_0^\infty \frac{\sin x \cos(ax)}{x} \, dx = \begin{cases} \frac{\pi}{2}, & |a| < 1, \\ \frac{\pi}{4}, & |a| = 1, \\ 0, & |a| > 1. \end{cases}

∫0∞sin⁡xx dx=∫0∞cos⁡xx dx=π2.\int_0^\infty \frac{\sin x}{\sqrt{x}} \, dx = \int_0^\infty \frac{\cos x}{\sqrt{x}} \, dx = \sqrt{\frac{\pi}{2}}.

2.4 Integrali con funzioni razionali

∫0∞xsin⁡(bx)a2+x2 dx=π2e−a∣b∣⋅sgn⁡(b).\int_0^\infty \frac{x \sin(bx)}{a^2 + x^2} \, dx = \frac{\pi}{2} e^{-a|b|} \cdot \operatorname{sgn}(b).

∫0∞cos⁡(ax)1+x2 dx=π2e−∣a∣.\int_0^\infty \frac{\cos(ax)}{1 + x^2} \, dx = \frac{\pi}{2} e^{-|a|}.

∫0∞sin⁡2(ax)x2 dx=π2∣a∣.\int_0^\infty \frac{\sin^2(ax)}{x^2} \, dx = \frac{\pi}{2} |a|.

2.5 Integrali di Fresnel

∫−∞∞sin⁡(x2) dx=∫−∞∞cos⁡(x2) dx=π2.\int_{-\infty}^\infty \sin(x^2) \, dx = \int_{-\infty}^\infty \cos(x^2) \, dx = \sqrt{\frac{\pi}{2}}.

2.6 Integrali ellittici

∫0π/2sin⁡x1−k2sin⁡2x dx=12kln⁡(1+k1−k),∣k∣<1.\int_0^{\pi/2} \frac{\sin x}{\sqrt{1 - k^2 \sin^2 x}} \, dx = \frac{1}{2k} \ln\left(\frac{1+k}{1-k}\right), \quad |k| < 1.

∫0π/2cos⁡x1−k2sin⁡2x dx=1karcsin⁡k,∣k∣<1.\int_0^{\pi/2} \frac{\cos x}{\sqrt{1 - k^2 \sin^2 x}} \, dx = \frac{1}{k} \arcsin k, \quad |k| < 1.

∫0π/2sin⁡2x1−k2sin⁡2x dx=1k2(K−E),∣k∣<1.\int_0^{\pi/2} \frac{\sin^2 x}{\sqrt{1 - k^2 \sin^2 x}} \, dx = \frac{1}{k^2} (K - E), \quad |k| < 1.

∫0π/2cos⁡2x1−k2sin⁡2x dx=1k2[E−(1−k2)K],∣k∣<1.\int_0^{\pi/2} \frac{\cos^2 x}{\sqrt{1 - k^2 \sin^2 x}} \, dx = \frac{1}{k^2} [E - (1-k^2)K], \quad |k| < 1.

Qui K=K(k)K = K(k) e E=E(k)E = E(k) sono gli integrali ellittici completi di prima e seconda specie.

∫0πcos⁡(nx)1−2bcos⁡x+b2 dx=πbn1−b2,∣b∣<1, n∈N0.\int_0^\pi \frac{\cos(nx)}{1 - 2b\cos x + b^2} \, dx = \frac{\pi b^n}{1-b^2}, \quad |b| < 1,\ n \in \mathbb{N}_0.

3. Integrali con funzioni logaritmiche

3.1 Integrali su [0,1]

∫01ln⁡(ln⁡x) dx=−γ.\int_0^1 \ln(\ln x) \, dx = -\gamma.

∫01ln⁡xx−1 dx=π26.\int_0^1 \frac{\ln x}{x-1} \, dx = \frac{\pi^2}{6}.

∫01ln⁡xx+1 dx=−π212.\int_0^1 \frac{\ln x}{x+1} \, dx = -\frac{\pi^2}{12}.

∫01ln⁡xx2−1 dx=−π28.\int_0^1 \frac{\ln x}{x^2-1} \, dx = -\frac{\pi^2}{8}.

∫01ln⁡(1+x)1+x2 dx=π8ln⁡2.\int_0^1 \frac{\ln(1+x)}{1+x^2} \, dx = \frac{\pi}{8} \ln 2.

∫01[ln⁡(1x)]a dx=Γ(a+1),a>−1.\int_0^1 \left[\ln\left(\frac{1}{x}\right)\right]^a \, dx = \Gamma(a+1), \quad a > -1.

3.2 Integrali trigonometrici con logaritmi

∫0π/2ln⁡(sin⁡x) dx=∫0π/2ln⁡(cos⁡x) dx=−π2ln⁡2.\int_0^{\pi/2} \ln(\sin x) \, dx = \int_0^{\pi/2} \ln(\cos x) \, dx = -\frac{\pi}{2} \ln 2.

∫0π/2xln⁡(sin⁡x) dx=−π28ln⁡2.\int_0^{\pi/2} x \ln(\sin x) \, dx = -\frac{\pi^2}{8} \ln 2.

∫0π/2sin⁡xln⁡(sin⁡x) dx=ln⁡2−1.\int_0^{\pi/2} \sin x \ln(\sin x) \, dx = \ln 2 - 1.

∫0πln⁡(a±bcos⁡x) dx=πln⁡(a+a2−b22),a≥b>0.\int_0^\pi \ln(a \pm b\cos x) \, dx = \pi \ln\left(\frac{a + \sqrt{a^2 - b^2}}{2}\right), \quad a \geq b > 0.

∫0πln⁡(a2−2abcos⁡x+b2) dx={2πln⁡a,a≥b>0,2πln⁡b,b≥a>0.\int_0^\pi \ln(a^2 - 2ab\cos x + b^2) \, dx = \begin{cases} 2\pi \ln a, & a \geq b > 0, \\ 2\pi \ln b, & b \geq a > 0. \end{cases}

∫0π/2ln⁡(tan⁡x) dx=0.\int_0^{\pi/2} \ln(\tan x) \, dx = 0.

∫0π/4ln⁡(1+tan⁡x) dx=π8ln⁡2.\int_0^{\pi/4} \ln(1+\tan x) \, dx = \frac{\pi}{8} \ln 2.

4. Integrali con funzioni algebriche

4.1 Integrali Beta

∫01xα(1−x)β dx=B(α+1,β+1)=Γ(α+1)Γ(β+1)Γ(α+β+2),\int_0^1 x^\alpha (1-x)^\beta \, dx = B(\alpha+1, \beta+1) = \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)},

valida per α>−1,β>−1\alpha > -1, \beta > -1.

4.2 Integrali con funzioni razionali

∫0∞dx(1+x)xa=πsin⁡(πa),0<a<1.\int_0^\infty \frac{dx}{(1+x)x^a} = \frac{\pi}{\sin(\pi a)}, \quad 0 < a < 1.

∫0∞dx(1−x)xa=−πcot⁡(πa),0<a<1.\int_0^\infty \frac{dx}{(1-x)x^a} = -\pi \cot(\pi a), \quad 0 < a < 1.

∫0∞xa−11+xb dx=πbsin⁡(πab),0<a<b.\int_0^\infty \frac{x^{a-1}}{1+x^b} \, dx = \frac{\pi}{b \sin\left(\frac{\pi a}{b}\right)}, \quad 0 < a < b.

4.3 Integrali vari

∫01dx1−xa=π Γ(1a)a Γ(1a+12).\int_0^1 \frac{dx}{\sqrt{1-x^a}} = \frac{\sqrt{\pi}\, \Gamma\left(\frac{1}{a}\right)}{a\, \Gamma\left(\frac{1}{a}+\frac{1}{2}\right)}.

∫01dx1+2xcos⁡a+x2=a2sin⁡a,0<a<π.\int_0^1 \frac{dx}{1+2x\cos a + x^2} = \frac{a}{2\sin a}, \quad 0 < a < \pi.

∫0∞dx1+2xcos⁡a+x2=asin⁡a,0<a<π.\int_0^\infty \frac{dx}{1+2x\cos a + x^2} = \frac{a}{\sin a}, \quad 0 < a < \pi. 

title: Note importanti
1. **Funzioni speciali:**
   - $\Gamma(x)$: funzione Gamma (integrale di Eulero di seconda specie).
   - $B(x,y)$: funzione Beta, $B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$.
   - $K(k), E(k)$: integrali ellittici completi di prima e seconda specie.
   - $\gamma$: costante di Eulero-Mascheroni $\approx 0,5772$).

2. **Condizioni di convergenza:** Tutte le formule includono esplicitamente le condizioni necessarie per la convergenza degli integrali.

3. **Relazioni tra integrali:** Alcuni integrali sono correlati (ad esempio, 25 con 9, 26 con 6, 27 con 7).

**Riferimenti:** E. Goursat, *Cours d'analyse mathématique*, Vol. II; tabelle standard di funzioni speciali.
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