It seems you've made a substitution that reversed what you did earlier:
The above is a pitfall to beware of when doing integration by parts.
I remember there was a similar thread a while back - see it in here.
I am having trouble with integrating by parts
I am trying to integrate e^2x * sin x
I have attached my working, and I am getting stuck towards the end. Please let me know where I am going wrong.
Here is the problem I was having when I tried to integrate witu u as e^2x :
And thanks for the link, I will look over it now
And here's as far as the OP originally got right, in balloon sculpture.
And filling the blanks (and solving the top row for I) does confirm his/her later answer, however arrived at.
... where (key in spoiler) ...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
Haha I have never seen anything like that before, Tom! I would love to learn how the baloons work tonight, but I will have to postpone it until exams are over.
I tried to use wolfram to confirm (1/3)*(e^(2x)*cos x +e^(2x) sin x)=(1/5)(2 sin x- cos x) but it wouldn't give the True or False answer it sometimes does.
Anyhow, thank you again TheCoffeeMachine and Tom@balooncalculus very much Good night
Ah okay thank you so much for taking the time to show me how to do this . One last thing, so should my second u and v' always be the same half of the equation if you know what I mean?
I let u= e^(2x) first but then for the second time I incorrectly let u=cos x instead of e^(2x) again... Is that what you meant I should be aware of when doing by parts?
Yes, basically making 'a substitution that precisely nullifies the effect of a given substitution'.One last thing, so should my second u and v' always be the same half of the equation if you know what I mean? I let u= e^(2x) first but then for the second time I incorrectly let u=cos x instead of e^(2x) again... Is that what you meant I should be aware of when doing by parts?