A Note From now onwards, we shall write only one constant of integration in the final answer. (ii) We have x INTEGRALS (iii) We have e – e dx – – log + C – log + C Example Find the following integrals: (sin cos ) (ii) cosec (cosec cot ) (iii) Solution We have (sin cos ) = – cos We have (cosec (cosec + cot ) cosec cosec cot = – cot cosec x + (iii) We have dx – x dx – = tan x + Example Find the anti derivative F of f defined by f ( x ) = x – , where F ( ) = Solution One anti derivative of f ( x ) is x – x since ) x – x = x – Therefore, the anti derivative F is given by F( x ) = x – x + C, where C is constant. Given that F( ) = , which gives, = – × + C or C = Hence, the required anti derivative is the unique function F defined by F( x ) = x – x + . Remarks We see that if F is an anti derivative of f , then so is F + C, where C is any constant.
Thus, if we know one anti derivative F of a function f , we can write down an infinite number of anti derivatives of f by adding any constant to F expressed by F( x ) + C, C ∈ R . In applications, it is often necessary to satisfy an additional condition which then determines a specific value of C giving unique anti derivative of the given function. Sometimes, F is not expressible in terms of elementary functions viz., polynomial, logarithmic, exponential, trigonometric functions and their inverses etc. We are therefore blocked for finding .
For example, it is not possible to find – x by inspection since we can not find a