Integral calculus
      The indefinite integral
      Integration by parts rule
      Recursion formulas - use of integration by parts formula
         Recursion formulas - use of integration by parts formula examples
Integration by parts rule
The rule for differentiating the product of two differentiable functions leads to the integration by parts formula.
Let  f (x) and g (x) are differentiable functions, then the product rule gives
[ f (x) g (x)]' =  f (x) g (x)'  +  g (x) f ' (x),
by integrating both sides
Since the integral of the derivative of a function is the function itself, then
and by rearranging obtained is
the integration by parts formula.
By substituting  u = f (x) and  v = g (x) then,  du = f ' (x) dx and  dv = g' (x) dx, so that 
To apply the above formula, the integrand of a given integral should represent the product of one function and the differential of the other.
The selection of the function u and the differential dv should simplify the evaluation of the remaining integral.
In some cases it will be necessary to apply the integration by parts repeatedly to obtain a simpler integral.
Recursion formulas - use of integration by parts formula
60.  Evaluate
Solution: 
Use the above recursion formula to calculate I2  and  I3
Example:  61.  Evaluate
Solution: 
Example:  62.  Evaluate
Solution: 
 63.  Evaluate
Solution: 
 
 64  Evaluate
Solution: 
   where the integral on the right side is of the type 3.
65  Evaluate
By using the substitution  x - x0 = t,  this integral leads to the type 3.
Solution: 
Use the above solution to evaluate the following example.
Example:  66.  Evaluate
Solution: 
67.   Evaluate
Solution: 
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