Integration by parts

INT [(ln x)/x]^2 dx
=INT -(ln x)^2 *(-1/x^2)dx
=INT -(ln x)^2 d(1/x)
= -(ln x)^2 * (1/x) - INT (1/x) d[ -(ln x)^2]
= -(ln x)^2 * (1/x) - INT (1/x) *[ (-2ln x) *(1/x) ]dx
= -(ln x)^2 * (1/x) +2 INT (ln x)(1/x^2) dx
= -(ln x)^2 * (1/x) -2 INT (ln x) [(-1/x^2) dx]
= -(ln x)^2 * (1/x) -2 INT (ln x) d(1/x)
= -(ln x)^2 * (1/x) -2 [(ln x)/x - INT (1/x) d(ln x)]
= -(ln x)^2 * (1/x) -2 (ln x)/x + 2 INT (1/x) (1/x) dx
= -(ln x)^2 * (1/x) -2 (ln x)/x + 2 INT (1/x^2) dx
= -(ln x)^2 * (1/x) -2 (ln x)/x + 2 (-1/x) + C
= -(ln x)^2 * (1/x) -2 (ln x)/x -2/x + C #

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