Integrating Function by Parts: Help Needed!

In summary: Yes, got it!In summary, the person tried to integrate by parts but ended up getting something wrong. They suggest making a substitution anddifferentiating to get the correct answer.
  • #1
greg_rack
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Homework Statement
$$\int \frac{lnx}{(x+1)^2} dx$$
Relevant Equations
none
Hi guys,

I've attempted to integrate this function by parts, which seemed to be the most appropriate method... but apparently, I'm getting something wrong since the result doesn't match the right one.
Everything looks good to me, but there must be something silly missing :)
My attempt:
Schermata 2021-03-02 alle 09.16.06.png
 
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  • #2
You seem to have gone wrong at the end. There is nothing you can do with ##\frac{\ln x}{x+1}##.
 
  • #3
PeroK said:
You seem to have gone wrong at the end. There is nothing you can do with ##\frac{\ln x}{x+1}##.
You mean I have gone wrong by collecting ##lnx## at the very end?
 
  • #4
greg_rack said:
You mean I have gone wrong by collecting ##lnx## at the very end?
Yes, the last three steps seem unnecessary and you've lost the term ##\ln|x +1|## somehow.
 
  • #5
PeroK said:
Yes, the last three steps seem unnecessary and you've lost the term ##\ln|x +1|## somehow.
Got it, that's actually taking me to nowhere... but still, I can't understand how to reach the solution
$$\frac{xlnx}{x+1}-ln(x+1)+c$$ provided by my textbook.
Is it just a simplification(that apparently I can't see) far from what I've got, or have I got something wrong in integrating?
 
  • #6
greg_rack said:
Got it, that's actually taking me to nowhere... but still, I can't understand how to reach the solution
$$\frac{xlnx}{x+1}-ln(x+1)+c$$ provided by my textbook.
Is it just a simplification(that apparently I can't see) far from what I've got, or have I got something wrong in integrating?
That combines the two terms you have in ##\ln x##.

PS Although it looks like they got the sign wrong.
 
  • #7
PeroK said:
That combines the two terms you have in ##\ln x##.
Uhm, but how? I could add the two terms in ##\ln x##, and still end up with a useless simplification
 
  • #8
greg_rack said:
Uhm, but how? I could add the two terms in ##\ln x##, and still end up with a useless simplification
The answer's wrong. They got the sign wrong on the parts, I suspect.
 
  • #9
PS You can always check an indefinite integral by differentiating, of course. That's something everyone should know!
 
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  • #10
PeroK said:
PS You can always check an indefinite integral by differentiating, of course. That's something everyone should know!
And indeed, differentiating my result, the integration appears to be incorrect, while the book's one is right... where did I get it wrong?🧐
 
  • #11
greg_rack said:
And indeed, differentiating my result, the integration appears to be incorrect, while the book's one is right... where did I get it wrong?🧐
You got the sign wrong, then! Right at the start.
 
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  • #12
Instead of integration by parts I suggest making the substitution ##u=\frac{1}{x+1}##, ##du=\frac{-dx}{(x+1)^2}##.
 
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  • #13
PeroK said:
You got the sign wrong, then! Right at the start.
Yes, got it!
Thanks a lot @PeroK :)
 

1. What is integration by parts?

Integration by parts is a method used to integrate a product of two functions. It is based on the product rule of differentiation and involves splitting the original integral into two parts and using a specific formula to solve it.

2. When is integration by parts used?

Integration by parts is used when the integrand (the function being integrated) is a product of two functions and cannot be solved using other integration techniques such as substitution or partial fractions.

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

When using integration by parts, the choice of which function to integrate and which function to differentiate is based on the acronym "LIATE". This stands for logarithmic, inverse trigonometric, algebraic, trigonometric, and exponential functions. The function that appears first in this list should be chosen as the one to integrate.

4. Can integration by parts be used for definite integrals?

Yes, integration by parts can be used for both indefinite and definite integrals. However, when using it for definite integrals, you may need to use the limits of integration to solve for the constant of integration.

5. Are there any common mistakes to avoid when using integration by parts?

Yes, some common mistakes to avoid when using integration by parts include forgetting to apply the formula correctly, choosing the wrong functions to integrate and differentiate, and not simplifying the resulting integral before solving for the constant of integration.

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