Indefinite Integrals


   
 
Some Special Types of Integrals
Following are few special integrals which can be integrated by using integration by parts
 
(i) Prove that
 
 
Proof:
 
 
 
Taking 1 as the second function and integrating by parts, we have
 
 
 
 
 
 
 
 
 
 
 
(ii). Prove that
 
 
Proof:
 
 
Taking 1 as the second function and integrating by parts, we have
 
 
 
 
 
 
 
 
 
 
 
 
 
(iii) Prove that
 
 
Proof:
 
Taking 1 as the second function
 
 
 
 
 
 
 
 
 
Integral of the form
 
 
Method:
 
The quadratic expression ax2 + bx + c can be expressed in the form a(x2  ± A2) by the method of completing the square. The integrals can be evaluated by using the special integrals.
 
Integrals of the form
 
 
Method:
 
 
 
(ii) Find the values of the constants L and M by comparing the co-efficients of like powers of x on both sides.
 
 
(iv) The integrals in the form are easily integrable.
 
Example:
 
 
Suggested answer:
 
 
 
 
Put x+1 = t
 
dx = dt
 
 
(using formula (iii))
 
 
  
     
   
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