Table of Integrals, Series, and Products, 8th Edition

3.24–3.27 Powers of x, of binomials of the form a + ßx^p and of polynomials in x

3.241

1. 

$\begin{array}{ll}{\displaystyle {\int }_0^1\frac{x^{\mu -1}\text{d}x}{1+x^p}}=\frac{1}{p}\beta \left(\frac{\mu }{p}\right)\hfill & \left[\mathrm{Re}\mu >0,p>0\right]\hfill \end{array}$

  WH, BI (2)(13)

2. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{x^{\mu -1}\text{d}x}{1+x}=\frac{\pi }{\nu }cos\text{}\text{ec}\frac{\mu \pi }{\nu }=\frac{1}{\nu }\text{B}\left(\frac{\mu }{\nu },\frac{\nu -\mu }{\nu }\right)}\hfill & \left[\mathrm{Re}\nu \ge \mathrm{Re}\mu \ge 0\right]\hfill \end{array}$

  ET I 309(15)a, BI (17)(10)

3.¹¹ 

$\begin{array}{ll}\text{PV}{\displaystyle {\int }_0^{\infty }\frac{x^{p-1}\text{d}x}{1-x^q}=\frac{\text{π}}{q}}\cot \frac{p\text{π}}{q}\hfill & \left[p<q\right]\hfill \end{array}$

  BI (17)(11)

4.¹² 

${\displaystyle {\int }_0^{\infty }\frac{x^{\mu -1}\text{d}x}{{\left(p+q{x}^{\nu }\right)}^{n+1}}=\frac{1}{\nu \text{}{p}^{n+1}}}{\left(\frac{p}{q}\right)}^{\mu /\nu }B\left(1+n-\frac{\mu }{\nu },\frac{\mu }{\nu }\right)$

  BI (17)(22)a

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