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A non-Jacobian numerical quadrature is proposed for evaluating improper integrals over a semi-infinite range. The quadrature first transforms the semi-infinite integration limit into a finite limit between -1 and 1. Standard numerical integration procedures such as Gauss-Chebyshev or Gauss-Legendre schemes can then be used to obtain the integral value. Unlike traditional methods using Laguerre or Hermite polynomials, numerical results show that no specific weight function is required for the proposed quadrature to converge as long as the integral exists. The transformation also includes a scale parameter which effectively expands the numerical-integration sampling points along its original semi-infinite integration path. Numerical examples shows that selection of the scale parameter is rather flexible and convergence can always be achieved within a wide range of parameter values. Several applied mechanics problems are also illustrated to validate the proposed numerical scheme. The quadrature can be easily incorporated into finite element or boundary element schemes for the evaluation of semi-infinite integrals.