Integrals By Parts With Infinity As Limit

[kloong]

$$\int_0^\infty \lambda x e^{-\lambda x} dx$$

How do I use the limits (infinity and 0) after getting the equation from integration by parts?

 

[Petr Mugver]

Just do the limits. Remember, if lambda > 0, polynomials dominate at x = 0, and exponentials dominate at infinity.

 

[Mark44]

You need to write your improper integral as the limit of a proper integral.
$$\int_0^\infty \lambda x e^{-\lambda x} dx = \lim_{b \to \infty} \int_0^b \lambda x e^{-\lambda x} dx$$

After you get your antiderivative, evaluate it at b and 0, and take the limit as b --> infinity.

 

[donifan]

No need to evaluate it. The integral is the first moment (mean) of an exponential distribution on x, so it is equal to $$\lambda^{-1}$$