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Integral calculus
  Integrating irrational functions
 Integration of irrational functions of the form
  Integrating irrational functions using Euler's substitutions
  Integrating irrational functions using Euler's substitutions examples
 Example:  Evaluate
As the leading coefficient of the quadratic a > 0 we can use first Euler's substitution, therefore
 Solution:

 Example:  Evaluate

As the leading coefficient a and vertical translation y0 of the quadratic have different signs, i.e., a · y0  < 0 the polynomial can be factorized using its real roots, hence we use second Euler's substitution,

 Solution:
 Example:  Evaluate
 Since the constant (or free) term of the given quadratic c > 0, we use third Euler's substitution, thus
 Solution:
  Binomial integral
Integral of the form
is called the binomial integral where, a and b are real numbers while m, n and p are rational numbers.
If  m, n and p all are integers then the integrand is a rational function integration of which is shown above.
There are only three cases the binomial integral can be solved by elementary functions:

1. if  m and  n are fractions and  p is an integer then, the integral can be solved using substitution x = t s, where s is the least common denominator of  m and n.

2.  if  p is a fraction and  (m + 1) / is an integer, then the integral can be solved using substitution         a + bxn = t s,  where  s  is denominator of  p.

3.  if  p is a fraction and  (m + 1) / n + is an integer then, the integral can be solved using substitution  ax - n + b = t s,  where  s  is denominator of  p.

 Solving binomial integrals examples
 Example:  Evaluate
 Solution:

 Example:  Evaluate
 Solution:

 Example:  Evaluate
 Solution:
 
 
 
 
 
 
 
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