Unit-III

Unit-III: Integral Calculus (09 Hrs.)

Contents:

Reduction Formulae, Beta and Gamma functions, Differentiation Under Integral Sign and Error functions

Introduction

Reduction Formula

: Reduction formulas are the formulas used to reduce the quantities, such as powers, percent, logarithms, and so. Reduction formulas are nothing but reducing an integral to another integral involving some variables such as x, n, t, and so on. ... For instance, the reduction formulae involving the variable n is denoted as I_n.

Gamma and Beta functions

This integral cannot be easily evaluated in general, therefore we first look at the Gamma function at two important points. ... Since at integer points, the value of the Gamma function is given by the factorial, it follows that the Gamma function grows to infinity even faster than exponentials.

Application of Gamma and Beta functions

While the gamma function behaves like a factorial for natural numbers (a discrete set), its extension to the positive real numbers (a continuous set) makes it useful for modeling situations involving continuous change, with important applications to calculus, differential equations, complex analysis, and statistics