# Integrating by Parts: Solving ∫r^3/(4+r^2)^(1/2) dr

• AnnieF
In summary, the conversation is about finding the integral of r^3/(4+r^2)^(1/2) using integration by parts or substitution. The experts suggest using u substitution with u=r^2+4, while others suggest using u=r^2 as it would make the integral easier to solve.

## Homework Statement

∫r^3/(4+r^2)^(1/2) dr

∫udv=uv-∫vdu

## The Attempt at a Solution

I know that integration by parts must be used. I tried doing it with 4+r^2 as u, but kept running into issues..then I got an answer but it appears to be wrong. I guess I am not sure how to do this.

First substitute r^2=t, then integrate by parts.ehild

I don't think it needs integration by parts. I'd do a simple u substitution. u=r^2+4, like AnnieF suggested. What 'issues' were you running into?

Dick said:
I don't think it needs integration by parts. I'd do a simple u substitution. u=r^2+4, like AnnieF suggested. What 'issues' were you running into?

The substitution $u=r^2$ would be significantly easier to deal with. One can then use parts or another substitution to make the integral elementary.

## 1. What is the formula for integrating by parts?

The formula for integrating by parts is ∫u dv = uv - ∫v du, where u and v are functions of x and dv and du are their respective derivatives.

## 2. How do I choose which function to assign as u and which as dv?

A common method is to choose u as the part of the integrand that becomes simpler after taking the derivative, and dv as the part that becomes more complicated after being integrated.

## 3. How do I solve the integral ∫r^3/(4+r^2)^(1/2) dr using integration by parts?

To solve this integral using integration by parts, you would assign u = r^3 and dv = (4+r^2)^(-1/2) dr, then use the formula ∫u dv = uv - ∫v du to find the solution.

## 4. What is the purpose of integration by parts?

The purpose of integration by parts is to simplify a complex integral by breaking it down into smaller, more manageable parts and applying the formula to find the solution.

## 5. Can integration by parts be used for any type of integral?

While integration by parts can be used for a wide range of integrals, it may not always be the most efficient method. It is important to consider other integration techniques, such as substitution or partial fractions, before deciding to use integration by parts.