Question

evaluate this question of beta Gamma function ​

Answer 1

Beta and Gamma Functions :

Question is having some error i.e., " Limits are running from 0 to 1 only "

Question :

$\; \displaystyle \sf \red{\int_{0}^{1} \left( \ln \dfrac{1}{x}\right)^{n-1}\ dx}$

Solution :

$\displaystyle \sf \int_{0}^{1} \left( \ln \dfrac{1}{x}\right)^{n-1}\ dx$

Let ,

$\sf u = \ln \dfrac{1}{x}$

⇒ u = - ln x

⇒ x = e⁻ᵘ

⇒ dx = - e⁻ᵘ du

Lower limit of u : ∞

Upper limit of u : 0

$\longrightarrow \displaystyle \sf \int_{\infty}^{0} \left( u \right)^{n-1}\ (- e^{-u}du)$

$\longrightarrow \displaystyle \sf - \int_{\infty}^{0} u^{n-1}\ e^{-u}\ du$

$\longrightarrow \displaystyle \sf \int_{0}^{\infty} u^{n-1}\ e^{-u}\ du$

$\bullet\ \; \orange{\displaystyle \sf \int_{0}^{\infty} x^{n-1}e^{-x}\ dx = \Gamma (n)}$

$\longrightarrow\ \displaystyle \sf \pink{\Gamma (n)}$

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