Could someone show the steps of how to find the anti derivative of using the integration by parts method
ln(2x+1)
See here: Integral ln(x)
First you need to rewrite this as $\displaystyle \displaystyle \frac{1}{2}\int{2\ln{(2x + 1)}\,dx}$ and make the substitution $\displaystyle \displaystyle u = 2x + 1 \implies du = 2\,dx$ to turn the integral into $\displaystyle \displaystyle \frac{1}{2}\int{\ln{(u)}\,du}$, and then follow the method in ASZ's link.
... might prefer [I would say]...
and if so, you may still find helpful to use...
... lazy integration by parts, doing without u and v.
... where (key in spoiler) ...
Spoiler:
However, for these 'log of a linear' type integrands, note the short cut of choosing an integral of 1 that has a constant of integration: i.e. fill out the rest of the product rule shape here...
... and then the expression to subtract on the bottom row (in order to keep the lower equals sign valid) is much simpler. (As per the link above.)
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Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
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I still don't see the method to everyone's answers
What I've done so far is set f = ln(2x+1) df = 2/(2x+1) dg = dx g = x
With that I get f * g - Integral(df * g)
Which equals x * ln(2x+1) - Integral[ 2/(2x+1) * x]
Using the same method on Integral[ 2/(2x+1) * x] I seem to go in circles with
f = x df = 1 dg = 2/(2x+1) g = ln(2x+1)
Instead should I make f = 2/(2x+1) df = -4/(2x+1)^2 dg = x g = x^2/2
In the second case it seems like my solution is getting more complicated but in the first one it goes in circles
Is the solution to just find a way to take the Integral[ 2/(2x+1) * x ] and simpify it in someway like someone above did using the method of making
2/(2x+1) * x = 1 - 1/(2x+1)
I guess what I really want to know is should I know that when my integration by parts method isnt working then it is time to play around with the form of the term so that it becomes easier to integrate just like doing
2/(2x+1) * x = 1 - 1/(2x+1)
It's not working because you don't integrate $\displaystyle \displaystyle \int{1 - \frac{1}{2x + 1}\,dx}$ by parts. Surely you know that the integral of $\displaystyle \displaystyle 1$ is $\displaystyle \displaystyle x$. The second term will give you a logarithm.
Of course, if you had followed my method of making a substitution in the first place, you wouldn't have this mess...
I wouldn't necessarily call it "dissing"; he does have a valid point on how "messy" it can get (its really not that bad, but an initial substitution makes things nicer [as Prove It suggested]). In that link you posted, we just happened to help out the user where they got stuck in their work. They practically set it up correctly, and we assisted them in finishing the problem instead of introducing "something new".
Of course, Prove It assumed he was dissing me, but he (and Mr F?) hadn't understood the OP because he hadn't followed the link.
Or, depending on the weather, we might think:
"a substitution is really not that bad, but going straight into parts is nicer, especially if we have the 'numerator give and take' method later on, or tom's shortcut".
Yes, but if you had shown a different approach as well (or several), let's hope no one would have lost their temper!
My Thanks to Prove It was for pointing out that making a simple substitution at the start made the problem 'cleaner'. My contextual understanding of the word 'mess' was as a metaphor directed at the OP, rather than a critical review of the help given by other people. See Laurel and Hardy: "Well here's another nice mess you've gotten me into, Stanley".
At any rate, I think this thread can be closed.
Edit: Sheesh ..... In light of obvious continued misunderstanding, the word "mess" used in this thread is clearly a metaphor for 'still having difficulties'. 'Nuff said.