用Γ函数表示下列积分:(1)∫(0,+∞)e^[-(x^n]dx(n>0)(2)∫(0,1)[ln(1/x)]^pdx(p>-1)(1)中0是下限,+∞是上限,答案是(1/n)*Γ(1/n)(2)中0是下限,1是上限,答案是Γ(p+1)求详解

Question

用Γ函数表示下列积分:(1)∫(0,+∞)e^[-(x^n]dx(n>0) (2)∫(0,1)[ln(1/x)]^pdx(p>-1)
(1)中0是下限,+∞是上限,答案是(1/n)*Γ(1/n)
(2)中0是下限,1是上限,答案是Γ(p+1)
求详解

Answer 1

(1)∫(0,+∞)e^[-(x^n]dx
令x^n=t x=t^(1/n) dx=(1/n)t^[(1/n)-1]dt
∫(0,+∞)e^[-(x^n]dx)=(1/n)∫(0,+∞)t^[(1/n)-1]e^(-t)dt=(1/n)*Γ(1/n)
(2)∫(0,1)[ln(1/x)]^pdx
令ln(1/x)=t x=e^(-t) dx=-e^(-t)dt x→0 t→+∞ x=1 t=0
∫(0,1)[ln(1/x)]^pdx=∫(+∞,0)t^p[-e^(-t)dt]=∫(0,+∞)t^[(p+1)-1]e^(-t)dt=Γ(p+1)