Integral calculus
The indefinite integral
The indefinite integrals containing quadratic polynomial (trinomial)
 55.  Evaluate
We solve given integral by separating the derivative of the quadratic polynomial in the numerator, thus
 solution:
therefore, the solution of I2 depend on the leading coefficient a and the vertical translation  y0, i.e.,
 a )   if then,
 b )  if then,
c )  if   y0 = 0  then,
 56.  Evaluate
 Solution:
The integrand function to be real, the quadratic polynomial in the denominator must be positive, therefore
a )   if  a > 0  and   y0 is not then
b )   if  a < 0  and   y0 > then
c )   if  a > 0  and   y0 = then
 57.  Evaluate

Use similar methods to solve this integral as in the preceding example. Hence, the solutions depend on the sign of the leading coefficient a and the sign or the value of the vertical translation y0.
 Example:  57 a )  Evaluate

Let solve given integral by separating the derivative of the quadratic polynomial in the numerator.
 Solution:
 58.  Evaluate
Use the substitution  mx + n = 1/ t .
 Example:  58 a )  Evaluate
 Solution:
 59.  Evaluate
 Solution:
The integrand function to be real, the quadratic polynomial must be positive, therefore
a )   if  a > 0  and   y0 is not then
see the solutions of the integrals, example 31 and 36. By using above substitutions
Note that the sign of the vertical translation y0 affects the solution, i.e., changes the sign of its second term.
b )   if  a < 0  and   y0 > then
see the solution of the integral example 8 above. After applying the substitutions
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