Unveiling Several Intricate Integrals with Inverse Tangent Function

Abstract

Calculus is the study of successive transformations in mathematics. Calculus includes two parts: differentiation and integration. Using the basic theorem of calculus, differentiation can be transformed into integration. Calculus use the infinite sequence and infinite series to converge to define the basic definition of limits. Differentiation can be simply understood as instantaneous change or just the slope of a function. The integration can be simply defined as calculate the area under the function. Integrals are divided into definite integrals and indefinite integrals. In the following, the paper mainly studies some specific definite integrals which related to inverse tangent function, polynomial, and logarithmic function. Through the method of substitution, integration by parts, exchange of integral order, and triangular substitution, all the answers have a rule, which is the integral related to π. Therefore, it is interesting to unveil the possible general solution of these integrals.