Integral calculus
      The indefinite integral
      The indefinite integrals containing quadratic polynomial (trinomial)
         The indefinite integrals containing quadratic polynomial, examples
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: 
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