# Improper Integral

• eestep

## Homework Statement

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

## The Attempt at a Solution

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

Use mathematical induction together with integration by parts.

eestep said:

## 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)!

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