3-ln-x-2-dx-

Question Number 91753 by Rio Michael last updated on 02/May/20

\int 3 {(\ln x)}^{2} d x = ? ?

Commented by peter frank last updated on 02/May/20

b y p a r t

Commented by mathmax by abdo last updated on 03/May/20

I = 3 \int {(l n x)}^{2} d x c h a n g e m e n t l n (x) = t g i v e

I = 3 \int t^{2} e^{t} d t =_{b y p s r t s} 3 {t^{2} e^{t} - \int 2 t e^{t}}

= 3 {t^{2} e^{t} - 2 ({t e}^{t} - \int e^{t} d t)}

= 3 {t^{2} e^{t} - 2 {t e}^{t} + 2 e^{t}) + c

= (3 t^{2} - 6 t + 6) e^{t} + c

= (3 {(l n x)}^{2} - 6 l n (x) + 6) x + c

= 3 x {(l n x)}^{2} - 6 x l n (x) + 6 x + c

Commented by Tony Lin last updated on 03/May/20

3 \int {(l n x)}^{n} d x

l e t t = l n x, \frac{d t}{d x} = \frac{1}{x} = \frac{1}{e^{t}}

3 \int e^{t} t^{n} d t

= 3 e^{t} \underset{k = 0}{\sum^{n}} {(t^{n})}^{(k)} {(- 1)}^{k}

= 3 x \underset{k = 0}{\sum^{n}} {(l n x)}^{k} P_{k}^{n} {(- 1)}^{k}, P_{k}^{n} = C_{k}^{n} \times k!

Answered by M±th+et+s last updated on 02/May/20

I = \int {l n}^{2} (x) d x

l e t u = {l n}^{2} (x) d v = d x

d u = \frac{2 l n (x)}{x} d x v = x

I = \int u d v

\int u d v = u v - \int v d u

= {x l n}^{2} (x) - \int x \frac{2 l n (x)}{x} d x

= {x l n}^{2} (x) - 2 \int l n (x) d x

= {x l n}^{2} (x) - 2 (x l n (x) - x) + c

\int 3 {l n}^{2} (x) d x = 3 I

Commented by Rio Michael last updated on 02/May/20

thanks sir

Commented by M±th+et+s last updated on 02/May/20

y o u a r e w e l c o m e