Integration by parts x5(lnx)2 dx

[LunarJK]

Homework Statement

Integrate the following:
(Ill use { as the integration sign)

{x⁵(lnx)² dx

Homework Equations

{u dv = uv - {v du

I'm really new to integration by parts, and unfortunately I am having to learn it out of a book for now. I sort of get the idea, but this one just doesn't look so right when I am done with it.

The Attempt at a Solution

u= (lnx)²
du = 2lnx(1/x) dx
v = x⁶/6
dv = x⁵dx

{u dv = (x⁶/6)((lnx)²) - { (x⁶/6)(2 lnx)(1/x)dx
= (x⁶/6)((lnx)²) - (1/3) { (x⁵)(lnx) dx

At this point I would doing integration by parts again for the integral: { (x⁵)(lnx) dx.

So:
u = lnx du = 1/x dx dv = x⁵dx v = x⁶/6
--> { (x⁵)(lnx) dx = (lnx)(x⁶/6) - { (x⁶/6)(1/x) dx
= (lnx)(x⁶/6) - (1/6) { x⁵ dx
= (lnx)(x⁶/6) - x⁶/12

Plugging this back into my original solution:

= (x⁶/6)((lnx)²) - (1/3)((lnx)(x⁶/6) - x⁶/12)
=(x⁶/6)((lnx)²) - (lnx x⁶/18) - x⁶/36This could even further be simplified i suppose by taking out some common factors of x⁶/6, but I don't even know if all this is right. Some let me know if I am on the right track, or where i went wrong??

 

[lanedance]

hi LunarJK

why not try and differentiate it and see if you get the original integrand?

 

[LunarJK]

hi LunarJK

why not try and differentiate it and see if you get the original integrand?

Well i tried... and i do not get the original integrand. I need some advice as to where i went wrong.

 

[lanedance]

your method looks good I think you may have just mixed up some +- signs and factors

i get to

[tex]= \frac{1}{6}(x^6(\ln{x})^2 - \frac{1}{3}( x^6\ln{x} -\frac{x^6}{6}) )+c = \frac{x^6}{6.6.3}(6.\ln{x)((3.\ln{x} - 1) + 1 ))+c[/tex]

which agrees with the online integrator
http://integrals.wolfram.com/

 

[lanedance]

Homework Statement

Integrate the following:
(Ill use { as the integration sign)

{x⁵(lnx)² dx

Homework Equations

{u dv = uv - {v du

I'm really new to integration by parts, and unfortunately I am having to learn it out of a book for now. I sort of get the idea, but this one just doesn't look so right when I am done with it.

The Attempt at a Solution

u= (lnx)²
du = 2lnx(1/x) dx
v = x⁶/6
dv = x⁵dx

{u dv = (x⁶/6)((lnx)²) - { (x⁶/6)(2 lnx)(1/x)dx
= (x⁶/6)((lnx)²) - (1/3) { (x⁵)(lnx) dx

this looks good here

At this point I would doing integration by parts again for the integral: { (x⁵)(lnx) dx.

So:
u = lnx du = 1/x dx dv = x⁵dx v = x⁶/6
--> { (x⁵)(lnx) dx = (lnx)(x⁶/6) - { (x⁶/6)(1/x) dx
= (lnx)(x⁶/6) - (1/6) { x⁵ dx
= (lnx)(x⁶/6) - x⁶/12

i think your 12 should be 6.6 = 36

Plugging this back into my original solution:

= (x⁶/6)((lnx)²) - (1/3)((lnx)(x⁶/6) - x⁶/12)


=(x⁶/6)((lnx)²) - (lnx x⁶/18) - x⁶/36

this should be -(-x^6/36) = +x^6/36 (i haven;t factored in the above error as well though...)

This could even further be simplified i suppose by taking out some common factors of x⁶/6, but I don't even know if all this is right. Some let me know if I am on the right track, or where i went wrong??

hopefully this gets you there