# How to Correctly Apply Integration by Parts to ln(x-7)?

• Monocles
In summary, the conversation is about solving an integration by parts question involving ln(x-7). The solution involves using u-substitution and integrating by parts to get (x-7)ln(x-7) - (x-7) + C as the answer. There is a discussion about the presence of +C in the answer and how it is an arbitrary constant. Mathematica also provides the same answer, but with the constant absorbed.
Monocles
[SOLVED] integration by parts question

## Homework Statement

This is part of a larger problem but I'm just not sure if I have the right answer.

edit: this is integral of ln(x-7)dx i just can't seem to figure out how to make it in latex

## The Attempt at a Solution

u-substituion:

$$u = x - 7$$

$$du = dx$$

Integration by parts to get

$$\intlnudu = ulnu - \int1du$$

$$= ulnu - u$$

$$= (x - 7)ln(x - 7) - (y - 7) + C$$

Now my only problem with this answer is that it gives 7 + C as part of the answer, and I can't recall ever seeing a constant + C as the answer for any integral.

I actually plugged the equation into this integral finder:

http://integrals.wolfram.com/index.jsp

And it told me that it was

$$(x-7)ln(x-7) - x$$

I'm guessing the +C is a given.

I would just leave it at +C and go on but its part of a differential equation so the value of C ends up being important...

Last edited:
C is not just a constant in the integration. It's an ARBITRARY constant. There is no difference between C and C+7. Just leave it at C and go on.

Alright thanks :)

You mean x-7 and a constant plus a constant is a constant. Both you and mathematica have the right answer, but mathematica just absorbed the 7 into the constant. Differentiate it to check. :)

## 1. What is integration by parts?

Integration by parts is a method used in calculus to find the integral of a product of two functions. It involves using the product rule for derivatives to rewrite the integral in a different form, making it easier to solve.

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

Integration by parts is typically used when the integral involves a product of two functions, or when the integrand cannot be easily integrated using other methods such as substitution or partial fractions.

## 3. How do I choose which function to differentiate and which to integrate?

The general rule is to choose the function that becomes simpler when differentiated, and the function that becomes more complex when integrated. This will usually involve choosing the function with the larger power of x as the one to differentiate.

## 4. What are some common examples of integration by parts?

Some common examples of integration by parts include finding the integral of e^x*sin(x) and ln(x).

## 5. How do I know if I've done the integration by parts correctly?

You can check if you've done the integration by parts correctly by differentiating your answer and seeing if it matches the original integrand. You can also compare your answer to the integral tables or use an online integral calculator for verification.

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