Another Integration By Parts Problem?

Question

Integral: (4x)(ln(5x))dx

Answer 1

u = ln(5x)
dv = 4x dx

It will work.

Answer 2

once you have an exponential circumstances something, consistently use the exponential simply by fact the 'v' function of the mixing with the aid of areas formulation (it is, yo evaluate exp(**)dx as a differential) Then regularly the subsequent vital to be achieved is far less complicated. in spite of the undeniable fact that with trig purposes, it would not substitute into much less complicated, that is yet another trig. in spite of the undeniable fact that, shop on a similar way, integrating the exponential and you gets the vital you began from with some coefficient (different from a million !) Then that grants you a linear equation in one unknown to your vital, it is amazingly merely got here upon. yet in a different way may be to place in writing a priori something like e^2x ( asinx + bcosx), differentiate this, hit upon a and b to retrieve you function to be integrated and that's it. First approach: Int e^2x sinx = a million/2e^2x sinx -a million/2Int e^2x cosx dx = a million/2e^2x sinx -a million/4e^2x cosx -a million/4Inte^2x sinx dx. for this reason: 5/4Int e^2x sinx = a million/2e^2x sinx -a million/4e^2x cosx 2d approach : 2e^2x(asin x + b cosx) + e^2x (acos x - bsin x) = e^2x((2a-b)sinx +(2b+a)cosx) for this reason 2a-b = a million and 2b+a = 0. b = -a/2 , 2a+a/2=a million, a = 2/5, b = -a million/5 and you spot which you get a similar answer as with the different approach.

Answer 3

4[ (ln(5x)) (x^2)/2 - integral {(1/x)[(x^2)/2]} ]

= 4[ (ln(5x)) (x^2)/2 - integral {x/2} ]

=4[ (ln(5x)) (x^2)/2 -(x^2) / 4 ]

= 2(ln(5x)) (x^2) - (x^2) +C

Answer 4

2x^2log(5x) - x^2