Question

I have a question about integration by parts. More specifically the integration of (x+a)lnx dx

Answer 1

looks like we need to integrate
\[x\ln(x)+a\ln(x)\]

Answer 2

the anti derivative of \(\ln(x)\) is \(x\ln(x)-x\)
you do this by parts, but it is so common a function that it is a lot easier to simply memorize this one, since you see the log all over the place

Answer 3

\(\int x\ln(x)dx\) you do by parts put \(u=\ln(x), du =\frac{1}{x}dx,dv=xdx,v=\frac{x^2}{2}\)

Answer 4

\[\int\limits_{1}^{e} (x+a) lnx dx\] is the problem but I'm not seeing where the varibale of a comes to play in this scheme.

Answer 5

you get
\[\int x\ln(x)dx=\frac{x^2}{2}\ln(x)-\frac{1}{2}\int xdx\] and you are almost home free

Answer 6

Where did the "a" go?

Answer 7

i took care of that in the first part
\[\int a\ln(x)dx=a(x\ln(x)-x)\]

Answer 8

so there is no "a" variable in the rest of the problem?