Volume 66 | Issue 1 | Year 2020 | Article Id. IJMTT-V66I1P511 | DOI : https://doi.org/10.14445/22315373/IJMTT-V66I1P511

The study aims to derive the generalized expression in the form of sequences and series for the nth indefinite integration of the following types of integrands: First, non-transcendental function with any rational number as an exponent; Second, the product of the non-transcendental and logarithmic function both having any rational number as an exponent; Third, the product of exponential, sine or cosine function with a non-transcendental function having any integral power. On analyzing the nth indefinite integration of type third integrand, it was observed that the solutions for the integral power of x can also be expressed in the form of Pascal’s triangle which revealed a divine beautiful symmetry by each row of Pascal’s triangle. In the proposed solution this symmetry generates a key for achieving the generalized expression for any higher integral exponent of x, that will help to study the relationship between the number theory and repeated integration of periodic functions, in which the coefficients of generalized expression can be expressed in terms of binomial coefficients and it can also be derived by rows of Pascal’s triangle. Interestingly, all these results are also written in an aesthetic form, which demonstrates the mathematical beauty in the generalized expressions which are discussed in this paper. In the application of fractional calculus, these derived expressions can be used to obtain the compact form of a particular integral of nth order differential equation with constant coefficients in just a single step. Here, the right-hand side of nth order differential equation is the integrand of the following forms of integrals: 1. 1/Dn = ∫∫∫.....∫(ax±b)±p/q(dx)n 2. 1/Dn = ∫∫∫.....∫(ax±p/q ±b)(dx)n 3. 1/Dn = ∫∫∫.....∫(ax±b)±p/q loge(ax±b)±r/s(dx)n 4. 1/Dn = ∫∫∫.....∫(ax±p/q ±b)loge(cx±r/s)(dx)n 5. 1/Dn = ∫∫∫.....∫(axm±b)e(cx±d)(dx)n 6. 1/Dn = ∫∫∫.....∫(axm±b) sin(cx±d)(dx)n 7. 1/Dn = ∫∫∫.....∫(axm±b) cos(cx±d)(dx)n

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M. Mebin Panamkuttiyiel Saumain, "Generalized Expression for the nth Integration of Transcendental and Non-Transcendental Functions," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 66, no. 1, pp. 61-91, 2020. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V66I1P511