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3.24–3.27 Powers of x, of binomials of the form a + ßxp and of polynomials in x

3.241

1.

WH, BI (2)(13)

2.

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

ET I 309(15)a, BI (17)(10)

3.11

$\begin{array}{ll}\text{PV}{\int }_{0}^{\infty }\frac{{x}^{p-1}\text{d}x}{1-{x}^{q}}=\frac{\text{π}}{q}\mathrm{cot}\frac{p\text{π}}{q}\hfill & \left[p

BI (17)(11)

4.12

${\int }_{0}^{\infty }\frac{{x}^{\mu -1}\text{d}x}{{\left(p+q{x}^{\nu }\right)}^{n+1}}=\frac{1}{\nu \text{​}{p}^{n+1}}{\left(\frac{p}{q}\right)}^{\mu /\nu }B\left(1+n-\frac{\mu }{\nu },\frac{\mu }{\nu }\right)$

BI (17)(22)a

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