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Definite integrals

Table of contents: Powers - Exponential functions - Sophomore's dream

Powers

463077
z1(ax+b)cdx=1a(c1)(az+b)c1\int_{z}^{\infty} \frac{1}{{\left(a x + b\right)}^{c}} \, dx = \frac{1}{a \left(c - 1\right) {\left(a z + b\right)}^{c - 1}}

Exponential functions

02e3d2
zeax+bdx=ebaza\int_{z}^{\infty} {e}^{-a x + b} \, dx = \frac{{e}^{b - a z}}{a}
9a06fb
zxceax+bdx=ebac+1Γ ⁣(c+1,az)\int_{z}^{\infty} {x}^{c} {e}^{-a x + b} \, dx = \frac{{e}^{b}}{{a}^{c + 1}} \Gamma\!\left(c + 1, a z\right)
f8de2e
zeax2+bdx=eb2πaerfc ⁣(az)\int_{z}^{\infty} {e}^{-a {x}^{2} + b} \, dx = \frac{{e}^{b}}{2} \sqrt{\frac{\pi}{a}} \operatorname{erfc}\!\left(\sqrt{a} z\right)
16a1f4
zeaxc+bdx=ebca1/cΓ ⁣(1c,azc)\int_{z}^{\infty} {e}^{-a {x}^{c} + b} \, dx = \frac{{e}^{b}}{c {a}^{1 / c}} \Gamma\!\left(\frac{1}{c}, a {z}^{c}\right)

Sophomore's dream

b77faf
01xxdx=n=1nn\int_{0}^{1} {x}^{-x} \, dx = \sum_{n=1}^{\infty} {n}^{-n}
66fefb
01xxdx=n=1(1)n+1nn\int_{0}^{1} {x}^{x} \, dx = \sum_{n=1}^{\infty} {\left(-1\right)}^{n + 1} {n}^{-n}

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2019-09-19 20:12:49.583742 UTC