`int x^5 ln(3x) dx` Find the indefinite integral

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Given to solve

`int x^5 ln(3x) dx`

let `u= ln(3x) ,u' = (ln(3x))'`

=`(1/(3x))*(3) = 1/x`

so `u' = 1/x`

and `v'= x^5 => v= x^6/6 `

by applying the integration by parts we get,

`int uv' dx= uv - int u'v dx`

so,

`int x^5 ln(3x) dx `

=`(ln(3x))(x^6/6) - int (1/x)(x^6 /6) dx`

= `(ln(3x))(x^6/6) - (1/6) int (1/x)(x^6 ) dx`

=` (ln(3x))(x^6/6) - (1/6) int (x^5 ) dx`

= `(ln(3x))(x^6/6) - (1/6) int (x^5 ) dx`

= `(ln(3x))(x^6/6) - (1/6) [x^6 /6]+c`

= `ln(3x)x^6/6 - 1/36 x^6 +c`

= `x^6/6(ln(3x)-x^6/6) + c`

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