function whose derivative is – x (iii) When the variable of integration is denoted by a variable other than x , the integral formulae are modified accordingly. For instance y y dy y EXERCISE . Find an anti derivative (or integral) of the following functions by the method of inspection. .
sin x . cos x . e x . ( ax + b ) .
sin x – e x Find the following integrals in Exercises to : . ( + ) . ( – . ax bx c dx .
. ( – x INTEGRALS . ( ) . ( 3cos .
( 3sin . (sec tan ) . cosec . – 3sin dx .
Choose the correct answer in Exercises and . . The anti derivative of equals (A) x (B) (C) x (D) . If d f x such that f ( ) = .
Then f ( x ) is (A) (B) (C) (D) . Methods of Integration In previous section, we discussed integrals of those functions which were readily obtainable from derivatives of some functions. It was based on inspection, i.e., on the search of a function F whose derivative is f which led us to the integral of f . However, this method, which depends on inspection, is not very suitable for many functions.
Hence, we need to develop additional techniques or methods for finding the integrals by reducing them into standard forms. Prominent among them are methods based on: . Integration by Substitution . Integration using Partial Fractions .
Integration by Parts . . Integration by substitution In this section, we consider the method of integration by substitution. The given integral can be transformed into another form by changing the independent variable x to t by substituting x = g ( t ).
Consider I = Put x = g ( t ) so that dx dt = g ′ ( t ). We write dx = g ′ (