Integral calculus
      The indefinite integral
      Integration by parts rule
      The indefinite integrals containing quadratic polynomial (trinomial)
         The indefinite integrals containing quadratic polynomial, 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.
The indefinite integrals containing quadratic polynomial (trinomial)
Recall that the quadratic polynomial  y = ax2 + bx + c  represents the expansion of the translatable form of
its source function  y = ax2  in the direction of the coordinate axes, thus
53.  Evaluate
 
 a )  Solution:   if  b2 - 4ac < 0  substitutions,
 
 
 b )  Solution:  if  b2 - 4ac > 0  substitutions,
 
 cSolution:  if  b2 - 4ac = 0  then  k = 0, substitution   gives
The indefinite integrals containing quadratic polynomial examples
Example:   53. a )   Evaluate
Solution: 
Example:   53. b )  Evaluate
 
Solution: 
Example:   53. c )  Evaluate
Solution: 
  54.  Evaluate
Since the numerator of the integrand is the derivative of the denominator therefore, the substitution
t = ax2 + bx + c  and   dt = (2ax + b) dx  gives
Solution:
 a )  Evaluate
by substituting in the numerator    
Solution:
 
Example:   54. a )  Evaluate
 
Solution: 
Functions contents G
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