Integrating irrational functions using Euler's substitutions

Integral calculus

The indefinite integral

Integrating irrational functions

Euler's substitutions

Integrals of the form

Euler's substitutions

Integrating irrational functions using Euler's substitutions examples

Example: 79. Evaluate

As the leading coefficient of the quadratic a > 0 we can use first Euler's substitution, therefore

solution:

Example: 80. Evaluate

As the leading coefficient a and the vertical translation y₀ of the given 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: 81. Evaluate

Since the constant (or free) term of the given quadratic c > 0, we use third Euler's substitution, thus

solution:

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