Inspired by the calculation formulas for infinite integrals containing trigonometric functions in the textbook "Mathematical Physics Methods", this paper mainly discusses the case where the integrand of such integrals has removable singularities on the real axis. Although this kind of problems can be solved using the integral contour, the method is quite complicated. Therefore, we replace the function integrated by another one which has no singular points on the real axis formally, and we use the formulas which results from the textbook to solve the integration whose function integrated has no singular points on the real axis formally, and then, according to the theory that the order can be exchanged for the integration and limitation under the certain conditions, we can obtain the results of the integration whose function integrated has removable singular points. Finally, the correctness of our method is proven through theoretical and practical examples.