integrate between the intervals negative infinity to positive infinity?

Question

[1/(1+2*pi^2*f^2)] df

integrate with respect to f

Answer 1

Use the substitution z = (π √2)f; then df = dz/(π √2). You get
1/(π √2) integral from -∞ to ∞ of dz/(1+z²)
= 1/(π √2) *atan(z) evaluated at -∞ and ∞
= 1/(π √2) [(π/2) - (-π/2)]

So the answer is 1/√2