Indefinite Integrals


   
 
Integration by Parts
Let u and v be two differentiable function of a single independent variable x.
 
 
 
Integrating w.r.t x on both sides
 
 
 
Let u = f(x),
 
Then
 
\ (1) can be written as
 
 
Note:
 
The above can be remembered as
 
 
(First function) (Integral of Second function)
 
 
Note:
 
(1) While integration by parts, the proper choice of first function and second function is significant
 
Try integrating
 
Taking x as the first function and ex as the second function is simpler than taking x as the second function.
 
(2) Integrating by parts may not be applicable to product of functions in all cases. In some cases the product of two functions may not be integrable.
 
(3) While finding the integral of the second function we do not add constant of integration.
 
We need not add a constant of integration to the second function as it gets cancelled in the final results.
 
(4) Some times even if the integral is not a product of two functions, the method of integration by parts can be used.
 
Example: Let us integrate sin-1x
 
 
 
 
 
where
 
 
 
 
 
Example:
 
Integrate the following function (x2 + 1) logx
 
Suggested answer:
 
Integrating by parts
 
 
 
 
 
 
 
Integral of the form
 
 
Method:
 
 
Put ex f(x) = t
 
Differentiating w.r.t. x,
 
{exf(x) +exf'(x)} dx = dt
 
ex{f(x) + f'(x)} dx = dt
 
= t + c
 
= exf(x)+c
 
Evaluate the integral .
 
Suggested answer:
 
 
 
 
 
 
Differentiating,
 
 
 
 
 
 
 
     
   
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