# Integration by Parts: Calculus Integral Help?

• MHB
• MarkFL
In summary, we are evaluating the integral $\int_1^2x^2\ln(x)\,dx$ and using integration by parts, we find that the answer is $\frac{8}{3}\ln(2)-\frac{7}{9}$.
MarkFL
Gold Member
MHB
Here is the question:

Calculus integral help?

Evaluate: the integral from 2(top) to 1(bottom) and the function is: x^2(lnx) dx

Here is a link to the question:

Calculus integral help? - Yahoo! Answers

I have posted a link there to this topic so the OP can find my response.

Hello Henry,

We are given to evaluate:

$\displaystyle \int_1^2x^2\ln(x)\,dx$

Using integration by parts, we may let:

$\displaystyle u=\ln(x)\,\therefore\,du=\frac{1}{x}\,dx$

$\displaystyle dv=x^2\,dx\,\therefore\,v=\frac{1}{3}x^3$

and so we have:

$\displaystyle \int_1^2x^2\ln(x)\,dx=\left[\frac{1}{3}x^3\ln(x) \right]_1^2-\frac{1}{3}\int_1^2x^2\,dx$

$\displaystyle \int_1^2x^2\ln(x)\,dx=\frac{8}{3}\ln(2)-\frac{1}{3}\left[\frac{1}{3}x^3 \right]_1^2$

$\displaystyle \int_1^2x^2\ln(x)\,dx=\frac{8}{3}\ln(2)-\frac{1}{9}\left(8-1 \right)$

$\displaystyle \int_1^2x^2\ln(x)\,dx=\frac{8}{3}\ln(2)-\frac{7}{9}$

## 1. What is integration by parts?

Integration by parts is a technique in calculus used to evaluate integrals of products of functions. It involves breaking down a complex integral into simpler parts and applying a specific formula to solve it.

## 2. When should I use integration by parts?

Integration by parts is useful when the integral involves a product of two functions, one of which can be differentiated and the other can be integrated. It is also used when other integration techniques, such as substitution, are not applicable.

## 3. How do I choose which function to differentiate and which to integrate when using integration by parts?

The general rule is to choose the function that becomes simpler after differentiating and the other function that becomes simpler after integrating. This choice may require some trial and error, but it usually becomes easier with practice.

## 4. What is the integration by parts formula?

The integration by parts formula is ∫u dv = uv − ∫v du, where u and v are the two functions in the integral and du and dv are their respective differentials. This formula can be used to evaluate integrals of products of functions.

## 5. Can integration by parts be used to solve definite integrals?

Yes, integration by parts can be used to solve both indefinite and definite integrals. When solving a definite integral using integration by parts, the integration by parts formula is applied first, and then the limits of integration are substituted into the resulting expression.

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