\int_0^\infty \sqrt{x}\,e^{-x}\,dx = \frac{1}{2}\sqrt \pi (see also Gamma function) \int_0^\infty e^{-a x^2}\,dx = \frac{1}{2} \sqrt \frac {\pi} {a} for a > 0 (the Gaussian integral) \int_0^\infty{x^2 e^{-a x^2}\,dx} = \frac{1}{4} \sqrt \frac {\pi} {a^3} for a > 0 \int_0^\infty x^{2n} e^{-a x^2}\,dx = \frac{2n-1}{2a} \int_0^\infty x^{2(n-1)} e^{-a x^2}\,dx = \frac{(2n-1)!!}{2^{n+1}} \sqrt{\frac{\pi}{a^{2n+1}}} = \frac{(2n)!}{n! 2^{2n+1}} \sqrt{\frac{\pi}{a^{2n+1}}} for a > 0, n is 1, 2, 3, ... and !! is the double factorial. \int_0^\infty{x^3 e^{-a x^2}\,dx} = \frac{1}{2 a^2} when a > 0 \int_0^\infty x^{2n+1} e^{-a x^2}\,dx = \frac {n} {a} \int_0^\infty x^{2n-1} e^{-a x^2}\,dx = \frac{n!}{2 a^{n+1}} for a > 0, n = 0, 1, 2, .... \int_0^\infty \frac{x}{e^x-1}\,dx = \frac{\pi^2}{6} (see also Bernoulli number) \int_0^\infty \frac{x^2}{e^x-1}\,dx = 2\zeta(3) \simeq 2.40 \int_0^\infty \frac{x^3}{e^x-1}\,dx = \frac{\pi^4}{15} \int_0^\infty \frac{\sin{x}}{x}\,dx = \frac{\pi}{2} (see sinc function and Sine integral) \int_0^\infty\frac{\sin^2{x}}{x^2}\,dx = \frac{\pi}{2} \int_0^\frac{\pi}{2}\sin^n{x}\,dx = \int_0^\frac{\pi}{2} \cos^n{x}\,dx = \frac{1 \cdot 3 \cdot 5 \cdot \cdots \cdot (n-1)}{2 \cdot 4 \cdot 6 \cdot \cdots \cdot n}\frac{\pi}{2} (if n is an even integer and n ≥ 2) \int_0^\frac{\pi}{2}\sin^n{x}\,dx = \int_0^\frac{\pi}{2}\cos^n{x}\,dx = \frac{2 \cdot 4 \cdot 6 \cdot \cdots \cdot (n-1)}{3 \cdot 5 \cdot 7 \cdot \cdots \cdot n} (if n is an odd integer and n ≥ 3) \int_{-\pi}^\pi \cos(\alpha x)\cos^n(\beta x) dx = \begin{cases} \frac{2 \pi}{2^n} \binom{n}{m} & |\alpha|= |\beta (2m-n)| \\ 0 & \text{otherwise} \end{cases} (for α , β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient) \int_{-\pi}^\pi \sin(\alpha x) \cos^n(\beta x) dx = 0 (for α , β real and n non-negative integer, see also Symmetry) \int_{-\pi}^\pi \sin(\alpha x) \sin^n(\beta x) dx = \begin{cases} (-1)^{(n+1)/2} (-1)^m \frac{2 \pi}{2^n} \binom{n}{m} & n \text{ odd},\ \alpha = \beta (2m-n) \\ 0 & \text{otherwise} \end{cases} (for α , β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient) \int_{-\pi}^{\pi} \cos(\alpha x) \sin^n(\beta x) dx = \begin{cases} (-1)^{n/2} (-1)^m \frac{2 \pi}{2^n} \binom{n}{m} & n \text{ even},\ |\alpha| = |\beta (2m-n)| \\ 0 & \text{otherwise} \end{cases} (for α , β, m, n integers with β ≠ 0 and m, n ≥ 0, see also Binomial coefficient) \int_{-\infty}^\infty e^{-(ax^2+bx+c)}\,dx = \sqrt{\frac{\pi}{a}}\exp\left[\frac{b^2-4ac}{4a}\right] (where exp[u] is the exponential function eu, and a > 0) \int_0^\infty x^{z-1}\,e^{-x}\,dx = \Gamma(z) (where \Gamma(z) is the Gamma function) \int_0^1 x^{\alpha-1}(1-x)^{\beta-1} dx = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} (for Re(α ) > 0 and Re(β) > 0, see Beta function) \int_0^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x) (where I0(x) is the modified Bessel function of the first kind) \int_0^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left(\sqrt{x^2 + y^2}\right) \int_{-\infty}^\infty (1 + x^2/\nu)^{-(\nu + 1)/2}\,dx = \frac { \sqrt{\nu \pi} \ \Gamma(\nu/2)} {\Gamma((\nu + 1)/2)} (for ν > 0 , this is related to the probability density function of the Student's t-distribution) If the function f has bounded variation on the interval [a,b], then the method of exhaustion provides a formula for the integral: \int_a^b{f(x)\,dx} = (b - a) \sum\limits_{n = 1}^\infty {\sum\limits_{m = 1}^{2^n - 1} {\left( { - 1} \right)^{m + 1} } } 2^{ - n} f(a + m\left( {b - a} \right)2^{-n} ). \int_0^1 \ln(1/x)^p\,dx = p!\; (Click "show" at right to see a proof or "hide" to hide it.)[show] The "sophomore's dream" \begin{align} \int_0^1 x^{-x}\,dx &= \sum_{n=1}^\infty n^{-n} &&(= 1.29128599706266\dots)\\ \int_0^1 x^x \,dx &= -\sum_{n=1}^\infty (-n)^{-n} &&(= 0.78343051071213\dots) \end{align} attributed to Johann Bernoulli. |
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