Integral calculus
      The indefinite integral
      Integration by parts rule
         Evaluating the indefinite integrals using the integration by parts formula, examples
         Evaluating the indefinite integrals using the integration by parts formula, solutions
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.
Evaluating the indefinite integrals using the integration by parts formula, examples
Evaluate the following indefinite integrals using the integration by parts formula;
47.   48.  
49.   50.  
51.   52.  
Evaluating the indefinite integrals using the integration by parts formula, solutions
Example:   47. Evaluate
Solution: 
Example:   48.  Evaluate
Solution: 
Example:   49.  Evaluate
Solution: 
Example:   50.  Evaluate
Solution: 
Example:   51.  Evaluate
Solution: 
Example:   52Evaluate
Solution: 
Calculus contents E
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