integration by parts --- integral x(ln x)^3 dx?

Question

u= x and dv = (lnx)^3


HELP!!!!!!!!

Answer 1

u= x and dv = (lnx)^3 won't work, because we don't know how to find the antiderivative of (lnx)^3. But we can make u = ln(x)^3 and and dv = x. Then, by parts,

integral x(ln x)^3 dx = x^2/2 . (lnx)^3 - Int (x^2/2) * 3 (lnx)^2 *1/x dx = x^2/2 . (lnx)^3 -3/2 - Int x (lnx)^2 dx

Now, we have reduced the grade of (lnx). If you continue in this way, you get to your integral. With some algebraic work, you can even find the integral of x (lnx)^n, where n>0 is an integer.

Answer 2

You have to choose to integrate the one that is easy to integrate.

Better let u = (lnx)^3
du = 3*(lnx)^2 / x

dv = xdx
v = x^2 / 2

In the next step you have to integrate x^2 * (lnx)^2

Apply the same rule again, then you'll have to integrate
x^3* lnx and so on.

Your final answer should be
x^2 * [ (1/2)*(lnx)^3 - (3/4)*(lnx)^2 + (3/4)*lnx - 3/8 ]

Answer 3

this will not work
as we do not know how to integrate (ln x)^3

try u = (ln x)^3 and dv = x dx
inegral
= uv - integral(x^2/2 (3) (ln x)^2)
= x^2/2(ln x)^3 -(3/2) integal(x^2(ln x)^2)

we have reduced the power of ln x from 3 to 2

apply repeatedly and get the result