# Integrating, probably by parts

• mbrmbrg
In summary, the conversation discusses different strategies for evaluating the integral \int{x(\ln{x})^3dx}. The person initially thought they had a quick way to integrate by parts, but ended up evaluating a different integral instead. They then revisited the original integral and attempted to use a substitution method, which resulted in a different answer. Eventually, they tried integration by parts with u = (ln(x))^3 and dv = x dx, which gave them the correct answer. They also mention that the substitution method can work, but it would require multiple iterations of integration by parts.
mbrmbrg
I have the expression $$\int{x(\ln{x})^3dx}$$
I thought I had a quick way to integrate by parts but it turned out that I had accidentally evaluated $$\int{x\ln{x}dx}$$ instead.
Revisiting $$\int{x(\ln{x})^3dx}$$, I wanted to start by making a strange substitution, wherein u=ln(x), du=1/x dx, and x=e^u. This meant that when I rewrote the integral, instead of multiplying dx by a constant to get it to be du, I multiplied it by x (which in this case was e^u). Is that allowed? Because I got a very different, much uglier answer than the book's.

I'd appreciate any comments, whether on my weird "method" or on a more standard approach to evaluating $$\int{x(\ln{x})^3dx}$$

Try integration by parts with u = (ln(x))^3 and dv = x dx

Last edited:
Your substitution method should work fine. Your should be integrating $$\int{Exp[2u] u^3du}$$. If you do it by integration by parts, you will need to do it 3 times.

wurth_skidder_23 said:
Try integration by parts with u = (ln(x))^3 and dv = x dx
thanks, that got me the book's answer!

And yes, the other way does work also. Nifty!

## What is integration by parts?

Integration by parts is a method used in calculus to simplify the integration of a product of two functions. It is based on the product rule for differentiation and is used to break down complex integrals into simpler ones.

## When should integration by parts be used?

Integration by parts should be used when the integral involves a product of two functions and it is not possible to use other integration techniques, such as substitution or partial fractions.

## How do you use integration by parts?

To use integration by parts, you need to identify the two functions in the integral and assign one as u and the other as dv. Then, you can use the formula ∫u dv = uv - ∫v du to simplify the integral.

## What is the formula for integration by parts?

The formula for integration by parts is ∫u dv = uv - ∫v du, where u and v represent the two functions in the integral and du and dv represent their respective differentials.

## Are there any limitations to integration by parts?

Yes, there are limitations to integration by parts. It can only be used for certain types of integrals and may not always result in a simpler integral. It also does not work for definite integrals with infinitely large or infinitely small limits.

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