Tricky integral: int [x^(2x)*(1+ln(x))] dx, from 2 to 4

[yossarian]

Alright I'm stumped

Integrate from 2 to 4

x^(2x)*(1+ln(x))*dx

I'm assuming u substitution is involved with this... along with screwing around with the properties of e and natural logs, but I have no clue where to even start. Thanks in advance.[/list]

 

[galactus]

Actually, there's an observation you can make to expedite things.

Remember the Second Fundamental Theorem of Calculus?.

Note, \(\displaystyle \L\\\frac{d}{dx}[x^{2x}]=2(ln(x)+1)x^{2x}\)

Therefore, \(\displaystyle \L\\\int{(ln(x)+1)x^{2x}dx=\frac{x^{2x}}{2}\)

I'll leave you evaluate using the limits.

 

[yossarian]

ah, thanks, was totally making it way more complicated than it needed to be.