Find the Limits of Integration for the Gamma Function

[eestep]

Homework Statement

Gamma function is defined for all x>0 by rule
\Gamma(x)=\int₀^\inftyt^x-1e^-tdt
Find a simple expression for \Gamma(n) for positive integers n. Answer is \Gamma(n)=(n-1)!

Homework Equations

The Attempt at a Solution

\int₀^\inftyt^n-1e^-tdt=-t^n-1e^-t-\int(n-1)t^n-2(-e^-t)dt=-t^n-1e^-t+(n-1)\intt^n-2e^-tdt
u=t^n-1 du=(n-1)t^n-2dt
dv=e^-tdt v=\inte^-tdt=-e^-t

 

[Metaleer]

Use mathematical induction together with integration by parts.

 

[eestep]

I appreciate the advice!

 

[SammyS]

Homework Statement

Gamma function is defined for all x>0 by rule
\Gamma(x)=\int_0^\infty\, t^{x-1}\,e^{-t}\,dt
Find a simple expression for \Gamma(n) for positive integers n. Answer is \Gamma(n)=(n-1)!

Homework Equations

The Attempt at a Solution

\Gamma(n)=\int_0^\infty\, t^{n-1}\,e^{-t}\,dt=-t^{n-1}\,e^{-t}-\int(n-1)t^{n-2}(-e^{-t})dt=-t^{n-1}e^{-t}\ +\ (n-1)\int t^{n-2}e^{-t}dt<br />

You are missing your limits of integration after doing integration by parts.

Click on the expression at the right to see the LaTeX code that produced it: \left[a^{-x}\right]_{\sqrt{2}}^{\infty}  .