Understanding the Derivative of ln^2: Integration by Parts

[a1ccook]

Homework Statement

I'm trying to understand how they got du in this problem. It is an integration by parts question. Anyway, the question asks me to evaluation this integral by using integration by parts two times

Homework Equations

This is how we would solve the problem...

The Attempt at a Solution

This is the first step.

The solution to the entire problem isn't what I need. I'll be able to get that after I figure out how they got du. I just need to see how you get du.

What it looks like they did was just go 2*(ln x^21)*(1/x)*(21x^2)
This isn't equal to the answer, though. Any ideas? No need to rush on this. The assignment is done, I just got monumentally frustrated with this one.

Thanks in advance to anyone that can help.

 

[annoymage]

you suppose to get

\frac{du}{dx} = 42(ln x²¹)x²⁰

 

[karkas]

Is the substitution really necessary, I mean, is it part of what you're being taught right now and must use it?

Because I find the following way to be more straightforward. Use integration by parts two times via this formula: \int f'(x)g(x)dx = f(x)g(x) - \int f(x)g'(x)dx

by doing the following: \int ln^2(x^{21})dx=\int (x)'ln^2(x^{21})dx=xln^2(x^{21}) - \int x(ln^2(x^{21}))'dx...

 

[annoymage]

but that what we learn at school, the substitution is merely to make student more understand i guess. But ones you get it, you don't need those substitution.

 

[D H]

you suppose to get

\frac{du}{dx} = 42(ln x²¹)x²⁰

Wrong. Look at the attachment to the original post. The correct answer is right there.So, to help with the original post.

[Anyway, the question asks me to evaluation this integral by using integration by parts two times

Did you do that?

This is the first step.

They gave you part of the first step. In particular, the attachment tells you exactly what to use how to express that integral in the form of \int u dv needed for application of the integration by parts. Then it tells you what du and v are given that assignment. It does not perform the actual integration by parts, that is up to you. Then, as suggested, you should integrate by parts again, obviously with a different choice for u and dv this second step.

 

[annoymage]

owho, sorry, i missed to differentiate something, hoho, sorry sorry