Methods of Integration

•    An integral is a mathematical object that can be interpreted as an area or a generalization of area.
•    The method of evaluating an integral by reducing it to standard form by a substitution is called integration by substitution.
•    Using substitution technique, we obtained the following standard integrals:

  • Integration of ‘tan x’ with respect to x is equal to log |sec x| + C
  • Integration of ‘cot x’ with respect to x is equal to log |sin x| + C
  • Integration of ‘sec x’ with respect to x is equal to log |sec x + tan x| + C
  • Integration of ‘cosec x’ with respect to x is equal to log |cosec x – cot x| +C

•    Any proper rational function f(x)/g(x) can be expressed as the sum of rational functions, each having a simple factor of g(x). Each such fraction is called a partial fraction. The process of obtaining partial fractions is called the resolution or the decomposition into partial fractions.

•    Integration by parts is an integration which is performed for product of two or more functions. In the method of integration by parts, we categorise two functions as the first function and the second function. Unlike differentiation where the order of functions does not matter, the order of choosing the first function and the second function for integration matters.
 

Algorithm to find Integration by Parts:
1.    Select the first function and the second function.
2.    If there is no other function, then take unity as the second function.
3.    Integrate the function by parts by using the following rule:
Integration of F1F2 = F1(Integration of F2) – integration of [(derivative of F1)(integration of F2)], where F1 and F2 are first and second functions respectively.
•    The ‘ILATE’ method is good for picking the first function (first in order of letters in ILATE, i.e., up in the table given below) so that it is easy to integrate. ILATE stands for:
I : inverse trigonometric functions: tan-1(x), sec-1 (x), etc.
L : logarithmic functions: ln(x), log2(x), etc.
A : algebraic functions: x2, 3x50, etc.
T : trigonometric functions: sin(x), tan(x), etc.
E : exponential functions: ex, 13x, etc.
•    Integration by parts is not applicable to the product of functions in all the cases.
•    While finding the integral of second function, we do not need to add any constant of integration.

Keywords: Integration by parts, Integration using trigonometric identities, Integration by substitution, Integration using partial fractions

 

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