Definite Integrals

By Shivendra Pratap|Updated : October 16th, 2021

Definite Integrals                                                   

                                                         

 

INTEGRATION

This is the inverse process of differentiation, if the differentiation of F(x) with respect to x be f(x) then the integration of f(x) with respect to x is F(x) i.e.,

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But the derivative of a constant term is zero then

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The process of finding the integral of a function is said to be integration and the function which is to be integrated is known as integrand.

Definite Integrals:

The definite integral is denoted by

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and is read as “the integral of the function f(x) w.r.t. ‘x’ from x = a to x = b”,

Property I.

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(i.e., the value of a definite integral depends on the limits and not on the variable of integration)

Property II

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(i.e., the interchange of limits changes the sign of the integral)

Property III.

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Property IV.

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Property V.

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Property VI.

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Wallis formula:

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Case-I. When n is odd,

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Reduction formula for 

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Here a generalized formula

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When m and n both are even K = π/2        

Otherwise K =1,

Leibnitz rule of Differentiation:

Let f(x, t) is integrand which is a function of two variables x and t then

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  1. Take care, here and are replaced in place of t in 2nd & 3rd term.
  2. If integrand is a function of ‘t’ alone then

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Gamma Functions:

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Beta function:

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