Table of Important Definite Integrals

1. Integrals with Exponential Functions

1.1 Integrals of the form $x^n e^{-ax}$

$$\int_0^\infty x^n e^{-ax} \, dx = \frac{\Gamma(n+1)}{a^{n+1}}, \quad a > 0,\ n > -1.$$

For positive integer $n$:

$$\int_0^\infty x^n e^{-ax} \, dx = \frac{n!}{a^{n+1}}.$$

$$\int_0^\infty x^n e^{-ax^2} \, dx = \frac{\Gamma\left(\frac{n+1}{2}\right)}{2a^{(n+1)/2}}, \quad a > 0,\ n > -1.$$

Particular cases:

• If $n = 2k$ (even):

  $$\int_0^\infty x^{2k} e^{-ax^2} \, dx = \frac{(2k-1)!! \sqrt{\pi}}{2^{k+1} a^{k+1/2}}.$$

• If $n = 2k+1$ (odd):

  $$\int_0^\infty x^{2k+1} e^{-ax^2} \, dx = \frac{k!}{2a^{k+1}}.$$

1.2 Gaussian Integrals

$$\int_0^\infty e^{-a^2 x^2} \, dx = \frac{\sqrt{\pi}}{2a}, \quad a > 0.$$

$$\int_0^\infty x^2 e^{-a^2 x^2} \, dx = \frac{\sqrt{\pi}}{4a^3}, \quad a > 0.$$

$$\int_0^\infty e^{-a^2 x^2} \cos(bx) \, dx = \frac{\sqrt{\pi}}{2a} e^{-b^2/(4a^2)}, \quad a > 0.$$

1.3 Integrals with Exponential Denominators

$$\int_0^\infty \frac{x}{e^x - 1} \, dx = \frac{\pi^2}{6}.$$

$$\int_0^\infty \frac{x}{e^x + 1} \, dx = \frac{\pi^2}{12}.$$

$$\int_0^\infty \frac{e^{-ax} \sin x}{x} \, dx = \arctan\left(\frac{1}{a}\right), \quad a > 0.$$

1.4 Integrals with Logarithms

$$\int_0^\infty e^{-x} \ln x \, dx = -\gamma,$$

where $\gamma \approx 0.5772$ is the Euler-Mascheroni constant.

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2. Integrals with Trigonometric Functions

2.1 Integrals of Powers of Sine and Cosine

$$\int_0^{\pi/2} \sin^{2\alpha+1} x \cos^{2\beta+1} x \, dx = \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{2\Gamma(\alpha+\beta+2)} = \frac{1}{2} B(\alpha+1, \beta+1),$$

valid for $\alpha > -1, \beta > -1$.

2.2 Integrals of the form $\frac{\sin(ax)}{x}$

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

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

2.3 Integrals with Trigonometric Combinations

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

$$\int_0^\infty \frac{\sin x \cos(ax)}{x} \, dx = \begin{cases} \frac{\pi}{2}, & |a| < 1, \\ \frac{\pi}{4}, & |a| = 1, \\ 0, & |a| > 1. \end{cases}$$

$$\int_0^\infty \frac{\sin x}{\sqrt{x}} \, dx = \int_0^\infty \frac{\cos x}{\sqrt{x}} \, dx = \sqrt{\frac{\pi}{2}}.$$

2.4 Integrals with Rational Functions

$$\int_0^\infty \frac{x \sin(bx)}{a^2 + x^2} \, dx = \frac{\pi}{2} e^{-a|b|} \cdot \operatorname{sgn}(b).$$

$$\int_0^\infty \frac{\cos(ax)}{1 + x^2} \, dx = \frac{\pi}{2} e^{-|a|}.$$

$$\int_0^\infty \frac{\sin^2(ax)}{x^2} \, dx = \frac{\pi}{2} |a|.$$

2.5 Fresnel Integrals

$$\int_{-\infty}^\infty \sin(x^2) \, dx = \int_{-\infty}^\infty \cos(x^2) \, dx = \sqrt{\frac{\pi}{2}}.$$

2.6 Elliptic Integrals

$$\int_0^{\pi/2} \frac{\sin x}{\sqrt{1 - k^2 \sin^2 x}} \, dx = \frac{1}{2k} \ln\left(\frac{1+k}{1-k}\right), \quad |k| < 1.$$

$$\int_0^{\pi/2} \frac{\cos x}{\sqrt{1 - k^2 \sin^2 x}} \, dx = \frac{1}{k} \arcsin k, \quad |k| < 1.$$

$$\int_0^{\pi/2} \frac{\sin^2 x}{\sqrt{1 - k^2 \sin^2 x}} \, dx = \frac{1}{k^2} (K - E), \quad |k| < 1.$$

$$\int_0^{\pi/2} \frac{\cos^2 x}{\sqrt{1 - k^2 \sin^2 x}} \, dx = \frac{1}{k^2} [E - (1-k^2)K], \quad |k| < 1.$$

Here $K = K(k)$ and $E = E(k)$ are the complete elliptic integrals of the first and second kind.

$$\int_0^\pi \frac{\cos(nx)}{1 - 2b\cos x + b^2} \, dx = \frac{\pi b^n}{1-b^2}, \quad |b| < 1,\ n \in \mathbb{N}_0.$$

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3. Integrals with Logarithmic Functions

3.1 Integrals on [0,1]

$$\int_0^1 \ln(\ln x) \, dx = -\gamma.$$

$$\int_0^1 \frac{\ln x}{x-1} \, dx = \frac{\pi^2}{6}.$$

$$\int_0^1 \frac{\ln x}{x+1} \, dx = -\frac{\pi^2}{12}.$$

$$\int_0^1 \frac{\ln x}{x^2-1} \, dx = -\frac{\pi^2}{8}.$$

$$\int_0^1 \frac{\ln(1+x)}{1+x^2} \, dx = \frac{\pi}{8} \ln 2.$$

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

3.2 Trigonometric Integrals with Logarithms

$$\int_0^{\pi/2} \ln(\sin x) \, dx = \int_0^{\pi/2} \ln(\cos x) \, dx = -\frac{\pi}{2} \ln 2.$$

$$\int_0^{\pi/2} x \ln(\sin x) \, dx = -\frac{\pi^2}{8} \ln 2.$$

$$\int_0^{\pi/2} \sin x \ln(\sin x) \, dx = \ln 2 - 1.$$

$$\int_0^\pi \ln(a \pm b\cos x) \, dx = \pi \ln\left(\frac{a + \sqrt{a^2 - b^2}}{2}\right), \quad a \geq b > 0.$$

$$\int_0^\pi \ln(a^2 - 2ab\cos x + b^2) \, dx = \begin{cases} 2\pi \ln a, & a \geq b > 0, \\ 2\pi \ln b, & b \geq a > 0. \end{cases}$$

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

$$\int_0^{\pi/4} \ln(1+\tan x) \, dx = \frac{\pi}{8} \ln 2.$$

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4. Integrals with Algebraic Functions

4.1 Beta Integrals

$$\int_0^1 x^\alpha (1-x)^\beta \, dx = B(\alpha+1, \beta+1) = \frac{\Gamma(\alpha+1)\Gamma(\beta+1)}{\Gamma(\alpha+\beta+2)},$$

valid for $\alpha > -1, \beta > -1$.

4.2 Integrals with Rational Functions

$$\int_0^\infty \frac{dx}{(1+x)x^a} = \frac{\pi}{\sin(\pi a)}, \quad 0 < a < 1.$$

$$\int_0^\infty \frac{dx}{(1-x)x^a} = -\pi \cot(\pi a), \quad 0 < a < 1.$$

$$\int_0^\infty \frac{x^{a-1}}{1+x^b} \, dx = \frac{\pi}{b \sin\left(\frac{\pi a}{b}\right)}, \quad 0 < a < b.$$

4.3 Various Integrals

$$\int_0^1 \frac{dx}{\sqrt{1-x^a}} = \frac{\sqrt{\pi}\, \Gamma\left(\frac{1}{a}\right)}{a\, \Gamma\left(\frac{1}{a}+\frac{1}{2}\right)}.$$

$$\int_0^1 \frac{dx}{1+2x\cos a + x^2} = \frac{a}{2\sin a}, \quad 0 < a < \pi.$$

$$\int_0^\infty \frac{dx}{1+2x\cos a + x^2} = \frac{a}{\sin a}, \quad 0 < a < \pi.$$

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