Mathematics

Session Indefinite Integrals -1

Session Objectives  Primitive or Antiderivative  Indefinite Integral  Standard Elementary Integrals  Fundamental Rules of Integration  Methods of Integration 1. Integration by Substitution, Integration Using Trigonometric Identities

Primitive or Antiderivative then the function F(x) is called a primitive or an antiderivative of a function f(x).

Cont. If a function f(x) possesses a primitive, then it possesses infinitely many primitives which can be expressed as F(x) + C, where C is an arbitrary constant.

Indefinite Integral Let f(x) be a function. Then collection of all its primitives is called indefinite integral of f(x) and is denoted by where F(x) + C is primitive of f(x) and C is an arbitrary constant known as ‘constant of integration’.

Cont. will have infinite number of values and hence it is called indefinite integral of f(x). If one integral of f(x) is F(x), then F(x) + C will be also an integral of f(x), where C is a constant.

Standard Elementary Integrals

Cont. The following formulas hold in their domain

Cont.

Fundamental Rules of Integration

Example - 1

Example - 2

Cont.

Example - 3

Example - 4

Integration by Substitution If g(x) is a differentiable function, then to evaluate integrals of the form We substitute g(x) = t and g’(x) dx = dt, then the given integral reduced to After evaluating this integral, we substitute back the value of t.

Cont.

Example - 5 Solution :

Integration Using Trigonometric Identities

Example - 6

Integration Using Trigonometric Identities

Example - 7 [Using 2sinAcosB = sin (A + B) + sin (A – B)]

Integration by Substitution

Example - 8

Solution Cont. Method - 2

Example - 9

Some Standard Results

Integration by Substitution

Example - 10

Integration by Substitution Use the following substitutions. (i) When power of sinx i.e. m is odd, put cos x = t, (ii) When power of cosx i.e. n is odd, put sinx = t, (iii) When m and n are both odd, put either sinx = t or cosx = t, (iv) When both m and n are even, use De’ Moivre’s theorem.

Example - 11 Powers of sin x and cos x are odd. Therefore, substitute sinx = t or cosx = t We should put cosx = t, because power of cosx is heigher

Cont.

Example - 12

Example - 13

Example - 14

Thank you