Integration of t/1-t2 dt

Question

For reduction in order in ordinary differential equations for assignment 

Answer 1

∫t/(1-t²) dt

 

Simple substitution works:

Let u = 1-t²

du = -2t dt, so du/(-2t) = dt

 

∫t/u du/(-2t) = -(1/2)∫1/u du = -(1/2)ln(u) + C = -(1/2)ln(1-t²) + C