Mathematics

Session Indefinite Integrals - 3

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

Three Standard Integrals

Integrals of the Form Reduce the given integral to one of the following forms:

Example-1

Example - 2

Solution Cont.

Example - 3

Integrals of the Form 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

Example - 4

Solution Cont.

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

Example - 5

Solution Cont.

Integration Through Partial Fractions (Type – 1) 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.

Example - 6

Solution Cont.

Type - 2 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.

Example - 7

Solution Cont.

Type - 3 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.

Example - 8

Solution Cont.

Type - 4 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

Example – 9

Solution Cont.

Example - 10 Solution: Here degree of N r > degree of D r.

Solution Cont.

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