# Integrating 3x^2*ln(x) - Is This Process Correct?

• feynmanpoint
In summary, the conversation discusses the process of integrating ∫3x^2*ln(x)=x^3*ln(x)-x^3/3+c, with the main topic being the step of factoring out x^3. The individual initially believed that this step was correct, but later realized their mistake.
feynmanpoint
I was working on a question and would this work?

∫3x^2*ln(x)

After I did all of the math, I got to:

∫3x^2*ln(x)=x^3*ln(x)-x^3/3+c

The problem I am having is not with the actual integration but another step I took:

∫3x^2*ln(x)=x^3*(ln(x)-1/3+c)

I figured that -1/3+c just makes another constant so I left it as:

x^3*(ln(x)+c)

and distributing x^3 renders:

x^3*ln(x)-c*x^3

Is this process correct?

No problem combining constants, HOWEVER - Why did you take that other step (from 2nd equation to third)? That is, why do you think that other step is correct?

TheoMcCloskey said:
No problem combining constants, HOWEVER - Why did you take that other step (from 2nd equation to third)? That is, why do you think that other step is correct?

I now realize I was wrong. I factored out an x^3 even though it didn't have one making my conclusion worthless.

## 1. What is the purpose of integrating 3x^2*ln(x)?

The purpose of integrating 3x^2*ln(x) is to find the antiderivative or the function that, when differentiated, gives 3x^2*ln(x) as the result. This is a fundamental concept in calculus and is used to solve various real-world problems.

## 2. How do you integrate 3x^2*ln(x)?

To integrate 3x^2*ln(x), you can use the integration by parts method. This involves breaking down the function into two parts and applying the product rule for differentiation in reverse. The resulting antiderivative will be in the form of a new function plus a constant.

## 3. Can you explain the steps involved in integrating 3x^2*ln(x)?

The steps for integrating 3x^2*ln(x) are as follows:

1. Apply the integration by parts formula: ∫u*dv = uv - ∫v*du
2. Choose u to be ln(x) and dv to be 3x^2, and find du and v.
3. Substitute the values into the formula: ∫ln(x)*3x^2 dx = 3x^2*ln(x) - ∫3x dx
4. Integrate the second term using the power rule: ∫3x dx = 3x^2/2 + C
5. Combine the terms to get the final result: ∫3x^2*ln(x) dx = 3x^2*ln(x) - 3x^2/2 + C

## 4. Is the process of integrating 3x^2*ln(x) reversible?

Yes, the process of integrating 3x^2*ln(x) is reversible. This means that if you take the antiderivative of the function, you will get back the original function 3x^2*ln(x). This is known as the Fundamental Theorem of Calculus.

## 5. Are there any other methods for integrating 3x^2*ln(x) besides integration by parts?

Yes, there are other methods for integrating 3x^2*ln(x). These include integration by substitution, integration by partial fractions, and integration by trigonometric substitution. However, integration by parts is the most commonly used method for this type of function.

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