A novel numerical integration method is presented, which uses the tangents at the beginning of the intervals in addition to the ordinates of the function to estimate the integral. The trapezoidal part of the area under the curve between tangent and abscissa is calculated exactly, while the area bounded between the curve and the tangent is estimated. This major reduction of the approximated part yields more accurate numerical integration results than current methods. The approximation is carried out using a mapped simple power function and a correction function. The method is tested using several functions including hard exponentials where most numerical integration methods fail to attain accurate results with limited numbers 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 these methods.