Integral calculus
The indefinite integral

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 I 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:
Calculus contents F