Integral calculus
The indefinite integral
The indefinite integrals containing quadratic polynomial (trinomial)
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,
 c )  Solution:  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:
Calculus contents F