integrate x/ln x dx?

Answer 1

Presumably what madhukar means is that this integral probably does not have a closed form solution. I agree - I don't see a good way to integrate it by parts.

You can find a power series for it by replace x=e^u and writing the intregral of:

e^(2u)/u du.

Then e^(2u) = sum (n=0,...,oo) (2u)^n/n!

= 1 + sum(n=1,...,oo) (2u)^n/n!

Dividing by u:

e^(2u)/u = 1/u + sum(n=1,oo) 2^n . u^(n-1)/n!

Integrating each term we get:

integral of e^(2u)/u = ln u + sum(n=1,oo) (2u)^n/(n.n!)

Putting u=ln x back into the equation, you get:

integral = ln ln x + sum (n=1,oo) (2 log x)^n/(n.n!)

Answer 2

i'm uncertain no count if that's ( a million/x)*ln x³ ... or ... a million / ( x. ln x³ ). consequently, I honestly have presented the two possible circumstances : ......................................... [ a million ] ? ( a million/x ). ln ( x³ ) dx = 3* ? ( ln x ) * ( a million/x ) dx ......(a million) ......................................... positioned u = ln x ... so as that ... du = ( a million/x ) dx. ......................................... From (a million), then, I = 3 * ? u du = 3 * ( u² / 2 ) + C = ( 3/2 )* ( ln x )² + C. ....................Ans. ......................................... [ 2 ] I = ? [ a million / ( x. ln ( x³ ) ) ] d = ? [ a million / ( 3* x. ln x ) ] dx = ( a million/3 ) * ? [ a million / ( ln x ) ] * ( a million/x ) dx = ( a million/3 ) * ? ( a million/u ) du, .......................the place u = ln x = ( a million/3 ) * ln u + C = ( a million/3 )* ln ( ln x ) + C. ............Ans. ......................................... chuffed to help ! .........................................

Answer 3

I doubt if it is integrable. Every function is not integrable.

Answer 4

(1.lnx - x. 1/x) / (ln x)^2

= ( ln x - 1) / ( ln x)^2

Answer 5

what are the bounds?