# Integrals By Parts With Infinity As Limit

• kloong
In summary, when using the limits (infinity and 0) after integration by parts, you need to write the improper integral as the limit of a proper integral. This can be done by evaluating the antiderivative at the limits and taking the limit as the upper limit approaches infinity. The integral represents the first moment (mean) of an exponential distribution on x, and is equal to \lambda^{-1}.
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?

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

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.

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

## 1. What is the concept of integrals by parts with infinity as limit?

Integrals by parts with infinity as limit is a mathematical technique used to evaluate integrals that involve products of functions. It involves using the formula ∫u dv = uv - ∫v du, where u and v are functions, and applying it repeatedly until the integral can be solved.

## 2. When is it necessary to use integration by parts with infinity as limit?

Integration by parts with infinity as limit is necessary when the integral involves a product of two functions, and one of the functions cannot be easily integrated. It is also used when evaluating integrals that have infinity as one of the limits.

## 3. How do you determine which function to choose as u and which to choose as dv?

The general rule for choosing u and dv is to choose u as the function that becomes simpler when differentiated, and dv as the function that becomes easier to integrate. However, there is no one correct way to choose u and dv, and sometimes trial and error may be needed.

## 4. Can integration by parts with infinity as limit be applied to definite integrals?

Yes, integration by parts with infinity as limit can be applied to definite integrals, as long as the limits of integration are not infinity. In this case, the final answer will be a finite number.

## 5. Are there any limitations to using integrals by parts with infinity as limit?

One limitation of integration by parts with infinity as limit is that it can only be used for integrals that involve products of functions. It also may not always provide a solution for the integral, and in some cases, other techniques such as substitution may be needed.

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