DEFINITE INTEGRAL | LEIBNITZ, REDUCTION FORMULA, INTEGRATION AS THE LIMIT OF SUM, PROPERTIES FOR PERIODIC FUNCTION | Reduction Formula: If `I_n = int_0 ^(pi/2) sin^nx = (n-1)/n I_(n-2)` and find `I_n` in terms of n for n being odd or even., If `I_(m, n) = int_0 ^(pi/2) sin^mx cos^nx dx` then show that `I_(m, n) = (m-1)/(m+n) I_(m-2, n)` and find `I_(m, n)` in terms of different combinations of m and n., INTEGRATION AS THE LIMIT OF SUM - Concept and physical interpretation, Evaluate the following integral as sum of Limits `int_1^3 (x^2+5x)`, Property 11: `int_0 ^(nT) f(x)= n int_0 ^T f(x) dx` where T is the period of the function and n is integer, Property 12: `int_a ^(a+nT) f(x) dx = n int_0 ^T f(x) dx` where T is period of f(x), Property 13: `int_(mT) ^(nT) f(x) dx = (n-m) int_0 ^T f(x) dx` where T is period of f(x) and m and n are integer, Property 14: `int_(a+nT) ^(b+nT) f(x) dx = int_a ^b f(x) dx` where T is the period of the f(x) and n is integer