76 dx x2 x2 36Solutionlet x 6tant dx 6sec2tdt I 766sec2

Solution

let x = 6tant
dx = 6sec^2tdt
I = 76*6sec^2tdt/36tan^2t(36tan^2t + 36)
I = 38/3 sec^2tdt/tan^2t*6sect
I = 19/9 sectdt/tan^2t
I = 19/9 cost/sin^2t dt
let sint = u
costdt = du
I = 19/9 du/u^2
integrating wrt u
I = -19/9u
I = -19/9sint
since tant = x/6
sint = x/(x^2+36)
hence
I = -19(x^2+36)/9x