Indefinite Integrals -1

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

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

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.

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’.

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.

The following formulas hold in their domain

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.

Solution :

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

Method - 2

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.

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

Indefinite Integrals - 2

 Integration by Parts  Integrals of the form

We express ax 2 + bx + c as one of the form x 2 + a 2 or x 2 – a 2 or a 2 – x 2 and then integrate.

We use the following method: (ii) Obtain the values of A and B by equating the like powers of x, on both sides. (iii) Replace px + q by A(2ax + b) + B in the given integral, and then integrate.

i.e. Integral of the product of two functions = First function x Integral of the second function – Integral of (derivative of first function x integral of the second function).

Proper Choice of First and Second Functions We can choose the first functions as the functions which comes first in the word ‘ILATE’, where I = Inverse trigonometric function L = Logarithmic function A = Algebraic function T = Trigonometric function E = Exponential function Note: Second function should be easily integrable.

[First Function = x, Second Function = cosx]

[Integrating by parts]

Indefinite Integrals - 3

 Three Standard Integrals  Integrals of the form  Integration Through Partial Fractions  Class Exercise

Reduce the given integral to one of the following forms:

We use the following method: (ii) Obtain the values of A and B by comparing the coefficients of like powers of x. Then the integral reduces to

We use the following method: (iii) Now, we evaluate the integral by the method discussed earlier.

When denominator is non-repeated linear factors where A, B, C are constants and can be calculated by equating the numerator on RHS to numerator on LHS and then substituting x = a, b, c,... or by comparing the coefficients of like powers of x.

When denominator is repeated linear factors where A, B, C, D, E and F are constants and value of the constants are calculated by substitution as in method (1) and remaining are obtained by comparing coefficients of equal powers of x on both sides.

When denominator is non-repeated quadratic factors where A, B, C are constants and are determined by either comparing coefficients of similar powers of x or as mentioned in method 1.

When denominator is repeated quadratic factors where A, B, C, D, E and F are constants and are determined by equating the like powers of x on both sides or giving values to x. Note: If a rational function contains only even powers of x, then we follow the following method: (i)Substitute x 2 = t (ii)Resolve into partial fractions (iii)Replace t by x 2

Solution: Here degree of N r > degree of D r.