# Intergration by Parts (IbP) problem

• Ravenatic20
In summary, the problem involves integrating x^2(ln x)^2 from 1 to 2e using integration by parts. After setting u = (ln x)^2 and dv = x^2 dx, and using the formula for integration by parts, we get an expression with a new integral on the right hand side. To solve this, we can use integration by parts again. The process is correct so far and the next step would be to integrate ln(x) again on the right hand side.
Ravenatic20
$$\int_{1} ^{2e} x^2(ln x)^{2} dx$$

I need to solve this using IbP. I made the following:

$$u = (ln x)^2$$

$$du = (\frac{2 ln x}{x}) dx$$

$$dv = x^2 dx$$

$$v = \frac{x^3}{3}$$

So I get:
$$(ln x)^2 (\frac{x^3}{3}) \|_{1} ^{2e}$$$$- \int_{1} ^{2e} (\frac{x^3}{3}) 2 ln x dx$$
(not sure how to make this look right)

Is this right?
Where do I go from here? Thanks

Integrate by parts again.

So what I have is right so far? I just IbP on the RHS?

Yup, if you've integrated ln(x) before, it should be obvious that you can integrate it again.

## What is Integration by Parts?

Integration by Parts (IbP) is a method used in calculus to find the integral of a product of two functions. It involves breaking the product into two parts and integrating them separately.

## When is Integration by Parts used?

Integration by Parts is typically used when the integrand is a product of two functions, or when the integrand cannot be solved by other methods such as substitution or partial fractions.

## What is the formula for Integration by Parts?

The general formula for Integration by Parts is ∫u dv = uv - ∫v du, where u and v are the two functions being integrated and dv and du are their respective differentials.

## How do you choose which function to use as u and which to use as dv?

When choosing u and dv, the acronym "LIATE" can be helpful: L - Logarithmic, I - Inverse Trigonometric, A - Algebraic, T - Trigonometric, E - Exponential. In general, u should be the function that becomes simpler after differentiation, and dv should be the function that can be easily integrated.

## What are some common mistakes to avoid when using Integration by Parts?

Some common mistakes to avoid when using Integration by Parts include forgetting to differentiate u or integrate dv, choosing the wrong functions for u and dv, and not simplifying the resulting integral. It is also important to be careful with signs and constants throughout the process.

• Calculus
Replies
8
Views
460
• Calculus
Replies
3
Views
561
• Calculus
Replies
4
Views
658
• Calculus
Replies
4
Views
1K
• Calculus
Replies
1
Views
2K
• Calculus
Replies
19
Views
3K
• Calculus
Replies
1
Views
281
• Calculus
Replies
2
Views
1K
• Calculus
Replies
1
Views
1K
• Calculus
Replies
6
Views
913