Improper Integral

[eestep]

Homework Statement

Gamma function is defined for all x>0 by rule
$$\Gamma$$(x)=$$\int$$₀^$$\infty$$t^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

$$\int$$₀^$$\infty$$t^n-1e^-tdt=-t^n-1e^-t-$$\int$$(n-1)t^n-2(-e^-t)dt=-t^n-1e^-t+(n-1)$$\int$$t^n-2e^-tdt
u=t^n-1 du=(n-1)t^n-2dt
dv=e^-tdt v=$$\int$$e^-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

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

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: $$\left[a^{-x}\right]_{\sqrt{2}}^{\infty}$$  .