Mathematics

Session Indefinite Integrals - 2

Session Objectives  Integration by Parts  Integrals of the form

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.

Example - 1

Example - 2

Solution Cont.

Example - 3

Solution Cont.

Example - 4

Example - 5

Solution Cont.

Example - 6

Integrals of the form 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.

Example – 7

Solution Cont.

Integration by Parts 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).

Integration by Parts (Cont.) 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.

Example - 8 [First Function = x, Second Function = cosx]

Example - 9

Solution Cont.

Example - 10 [Integrating by parts]

Solution Cont.

Integrals of the form

Example - 11

Solution Cont.

Integrals of the form

Example - 12 [Integrating by parts]

Solution Cont.

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