In this study, a proposed numerical integration method is presented. This novel method uses the tangents at the beginning of the intervals in addition to the ordinates of the function to estimate the integral numerically. Trapezoidal part of the area under the curve between tangent and abscissa is calculated exactly, while area bounded between the curve and the tangent is estimated. This major reduction of the approximated part, on one hand, yields more accurate numerical integration results. On the other hand, the approximation is carried out using a mapped simple power function and a proper correction accordingly. The method is tested using several functions including hard exponentials where most numerical integration methods fail to attain accurate results with limited number of intervals, since the ordinates of such functions increased rapidly within a short boundary region. The proposed method is compared with the well-known Simpson’s 1/3 Rule, Gauss Quadrature, in addition to the MATLAB quad function. The numerical results demonstrate that the proposed method is superior to the mentioned methods. Used the proposed method in several real-world applications that would benefit from having the boundary layer problem solved. Here had been studied three structural mechanics applications (beam with variable cross-sections with different loading cases and different varying types, Arch with variable cross-sections with different loading cases and different varying types). The numerical results demonstrate that the proposed method is superior to the mentioned methods.