Integrate 1 / ((x^5) * sqrt (9*x^2-1)) ... ?

Let 9x^2-1 = y^2

=> 18xdx = 2ydy

=> ydy = 9xdx

lower limit = sqrt (9*2/9 - 1) = sqrt (1) = 1

upper limit = sqrt (9*4/9 - 1) = sqrt (3)

Int. [sqrt (2) / 3,2/3] 1 / (x^5 (sqrt (9x^2-1)) dx

= Int. [sqrt (2) / 3,2/3] xdx / (x^6 (sqrt (9x^2-1))

= 81 * Int. [1, sqrt (3) ] ydy / ((y^2+1) ^3y)

=81 * Int. [1, sqrt (3) ] dy / (y^2+1) ^3

y=tanz

dy = sec^2z dz

=81*Int [pi/4, pi/3] cos^4 (z) dz

=81/4*int [pi/4, pi/3] (1+cos (2z)) ^2 dz

=81/4 * Int. [pi/4, pi/3] (1+2cos (2z) + cos^2 (2z)) dz

=81/4 * (pi/3-pi/4) + 81/4 * (sin (2pi/3) - sin (pi/2)) + 81/8 * (pi/3-pi/4)

+ 81/32 * (sin (-pi/3) - sin (pi))

=81 (pi/48+pi/96+1/4 * (sqrt (3) / 2 - 1) - 1/32 * sqrt (3) / 2)

=81/32 * (pi+3sqrt (3) - 8)