# How Do You Solve Integration by Parts for ∫ x*(ln(x))^4 dx?

• fallen186
In summary, the conversation is about integration by parts and the correct choice of substitution in a particular problem. The participants discuss the use of a chart method and the importance of choosing the correct substitution, with the suggestion to prioritize logarithmic terms as the u substitution. They also mention the possibility of using reduction formulas and the use of an acronym for remembering the order of priority for choosing u.
fallen186

## Homework Statement

$$\int x*(ln(x))^4dx = 4ln|x|^3-12ln|x|$$

## The Attempt at a Solution

I did chart method
u...dv...+/-
------------------------
x...ln|x|^4...+
1...(4ln|x|^3)/x.. -
0...12ln|x|...+
......-

What's the question?
What's the significance of your chart? Is u = x? And is dv = (ln|x|)^4 *dx? If so, this is not at all a useful substitution.

The chart is the authors attempt at a Tabular Integration by Parts: http://en.wikipedia.org/wiki/Integration_by_parts#Tabular_integration_by_parts

Hes probably asking us just to check his work.

To fallen186 - When we do Tabular integration, the Column with our chosen "u" is the derivatives column, whilst the "dv" column are for the integrals, not for derivatives again. And as Mark44 said, its probably better if you reconsider your substitutions.

I was taught this about picking the correct substitution:

HIGHEST priority for choice of u (i.e. make these things "u")
lnx logx arctanx and things like that LEVEL 1
x^2 (i.e. polynomials) or things like that LEVEL 2
cos x sin x LEVEL 3
e^x LEVEL 4
LOWEST priority of choice of u.

Have you done such?

Also, many textbooks include reduction formulas in the back and sometimes test just to see if you are familiar with such formulas. This may be the case here as well, although using integration by parts is not at all difficult in this problem.

Yes I have, the algorithim for picking the substitution is often shortened to "ILATE" or "LIATE". I learned ILATE but it doesn't really matter unless you have some nasty product of both an Inverse Trig and a log.

Basically were saying, fallen186, you should pick your log term as your u sub.

## 1. What is Integration by Parts?

Integration by Parts is a mathematical technique used to find the integral of a product of two functions. It is a way to break down a complex integral into simpler parts that can be more easily solved.

## 2. When should I use Integration by Parts?

Integration by Parts is typically used when the integrand (the function being integrated) contains a product of functions that are difficult to integrate directly. It is also useful when the integrand contains a combination of polynomials, trigonometric functions, and exponential functions.

## 3. How do I use Integration by Parts?

To use Integration by Parts, you must first identify which part of the integral can be differentiated and which part can be integrated. This is typically done by using the acronym "LIATE" which stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential. Once you have identified the parts, you can apply the Integration by Parts formula to solve the integral.

## 4. What is the Integration by Parts formula?

The Integration by Parts formula is ∫u dv = uv - ∫v du. This formula can be remembered using the mnemonic "u-du, v-dv". In this formula, u represents the function that will be differentiated, and dv represents the function that will be integrated.

## 5. Are there any tips or tricks for using Integration by Parts?

One helpful tip for using Integration by Parts is to choose u and dv in a way that will simplify the integral as much as possible. Another tip is to choose u such that it becomes simpler after being differentiated, and dv such that it becomes simpler after being integrated. Additionally, it is important to practice and become familiar with the different types of integrals that can be solved using Integration by Parts.

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