Improper integration on a line

Stream: Is there code for X?

[Yuval Filmus]

Chebshev polynomials are orthogonal with respect to the weight function 1/sqrt(1-x^2) (supported on (-1,1)). The proof goes by a substitution. I tried writing out the proof using the standard integral on a line, but I need continuity of the integrand on the half-open interval, whereas here it is continuous only on the open interval (-1,1).
How to proceed?

[Weiyi Wang]

Does it work if you cut it in half and do (-1, 0] and [0,1) separately?

[Jireh Loreaux]

@Yuval Filmus if you provide a #mwe, you're much more likely to have help getting unstuck.