For u = t2/7, guide on the formula d[arctan (t2/7)]/dt = (2t/7)/{1 + (t2/7)2}*.
Write ∫{t dt/[t4 + 49]} as 0.5∫{2t dt/[(t2)2 + 49]}.
Next express the integral as 0.5∫{(2t/49)dt/[(t2)2/49 + 49/49]}.
Rewrite as (1/7)(1/2)∫7{(2t/49)dt/[(t2/7)2 + 1]} which goes to
(1/14)∫{(2t/7)dt/[(t2/7)2 + 1]} or 14-1 arctan [t2/7] + C.
*For u = t2/7, du/dt = 2t/7. Also, y = arctan u, tan y = u. Implicit Differentiation
of tan y = u gives sec2y(dy/dt) = du/dt or 2t/7 and dy/dt = (2t/7)/sec2y or
(2t/7)/(1/cos2y) which translates to (2t/7)/[(cos2y + sin2y)/cos2y] or
(2t/7)/[(1 + tan2y)] or (2t/7)/{1 + (t2/7)2}.
