Question

How to calculate or simplify the the integral of ((18(ln(x))^5)/x)dx ?

Answer 1

Then,
the property of ln() is that if it's raised to anything, you can take that out in front:
ln(x)^2 = 2ln(x)

Take it from there...

Answer 2

by parts maybe? let u = ln(x) and dv = 1

Answer 3

\[\int\limits \frac{ 18\left( \ln \left( x \right) \right)^5 }{ x }dx\]

Answer 4

like this?

Answer 5

theres an x under there eh .... then just a u sub seems sufficient

Answer 6

if so, let u = ln(x), du = (1/x) dx

Answer 7

Yes pgpilot326 ! thank you

Answer 8

And how do you integrate an ln function? I can't find a clear definition of it in my book.

Answer 9

\[ln^5(x)\ne ln(x^5)\]
so that exponent property doesnt fit

Answer 10

the key to this is in knowing how to take the derivative of the ln function

Answer 11

in this one you don't have to integrate ln(x)

Answer 12

when that becomes clear, it can be seen that the chainrule was applied and it cleans up with the u subsititution

Answer 13

I think I got it. Does it become 18(u)^5 du?

Answer 14

it does :)

Answer 15

Ok thank you all very much!

Answer 16

\[\int\limits \frac{ 18\left( \ln \left( x \right) \right)^5 }{ x }dx=18\int\limits u^5 du=\frac{ 18u^6 }{ 6 }=3u^6 = 3\left( \ln \left( x \right) \right)^6\]

Answer 17

waiting for someone to shime in with the +C

Answer 18

i'm singing it... 1 octave above middle C