How to evaluate conditional expression for definite integral [closed]

Question

I am trying to show that the definite integral

Integrate[
  (y*(1 - y) + (1 - t)*y^2 + t*(y - 1)^2)/((1 - t)*y^2 + t*(y - 1)^2)^2,
  {y, -Infinity, Infinity}
]

is

$$\frac{\pi}{\sqrt{t(1-t)}}$$

with $t$ restricted to be real. However, the code above returns a conditional expression. Subsequently simplifying to restrict $t$ to be real returns 'undefined'. What am I doing wrong?

Answer 1

If you want to give conditions to an integrand, you may use the option "Assumptions->...". In your case:

Integrate[(y*(1 - y) + (1 - t)*y^2 + 
    t*(y - 1)^2)/((1 - t)*y^2 + t*(y - 1)^2)^2, {y, -Infinity, 
  Infinity}, Assumptions -> t \[Element] Reals]