# Simple Integral Help: Evaluate x^2*ln(x)dx

• ebk11
In summary, the integral of x^2 isn't 2x, and if dv = x^2, then to get v, you just need the integral of x^2, which is just 2x.
ebk11

## Homework Statement

Evaluate the following integral:

[integral](x^2)*ln(x)dx

## Homework Equations

I believe this is substitution by parts...

## The Attempt at a Solution

I chose u = ln(x), and dv = x^2. The problem I am having is I can't figure out what du and v should be, because I am thinking too much and confusing myself. If u = ln(x), then du should simply be 1/x, right? And if dv = x^2, then to get v, you just need the integral of x^2, which is just 2x. Is that right?

If those are right, then how come I come up with the following answer, which I don't believe is right (and I will laugh if it is but I really don't think so):

2x*lnx - 2x + C

Last edited:
The INTEGRAL of x^2 isn't 2x.

ebk11 said:
And if dv = x^2, then to get v, you just need the integral of x^2, which is just 2x. Is that right?

No. You have differentiated, not integrated.

Well that's what I was confused about.

When you are setting up a substitution by parts problem...when you get u, you differentiate to get the du...and then you integrate to get v from dv, right?

So it should be x^3/3, not 2x?

ebk11 said:
So it should be x^3/3, not 2x?

Correct. Technically speaking you'd have:

u = ln x
dv = x2 dx

fss said:
Correct. Technically speaking you'd have:

u = ln x
dv = x2 dx

Okay, so I got du = 1/xdx, and v = x^3/3

Then, I did: lnx * x^3/3 - [integral]x^3/3 * 1/xdx

then: lnx * 1/3*x^3 - 1/3[integral]x^3/xdx

which simplifies to: lnx * 1/3 * x^3 - 1/3[integral]x^2dx

which is: lnx * 1/3 * x^3 - 1/3[integral]x^3/3 + C

final answer: ln(x) * 1/3 * x^3 - 1/9 * x^3 + C

Does that seem right?

Looks like you got it to me.

## 1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to find the total value of a continuously changing quantity, such as velocity or volume, over a given interval.

## 2. How do you evaluate an integral?

To evaluate an integral, you need to use a specific method, such as the power rule, substitution, or integration by parts. These methods involve breaking down the integral into smaller, simpler parts and using mathematical rules to solve it.

## 3. What is the power rule?

The power rule is a method for evaluating integrals of the form x^n, where n is any real number. It states that the integral of x^n is equal to (x^(n+1))/(n+1) + C, where C is a constant. In other words, you add 1 to the exponent, divide by the new exponent, and add a constant to the result.

## 4. What is substitution in integration?

Substitution is a method for evaluating integrals that involves replacing a variable in the integrand (the expression being integrated) with a new variable. This new variable is chosen so that it simplifies the integral and makes it easier to solve. After integrating, the new variable is replaced with the original variable.

## 5. How do you evaluate the integral x^2*ln(x)dx?

To evaluate this integral, you can use the integration by parts method. This involves breaking down the integral into two parts, one of which is differentiated and the other is integrated. The resulting equation can then be solved for the original integral. In this case, you would let u = ln(x) and dv = x^2 dx. After applying the integration by parts formula, you would get the final answer of (x^2)(ln(x) - 1) + C.

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