Recursion formulas and use of integration by parts formula

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Integral calculus

The indefinite integrals containing quadratic polynomial of the form

Solution:

The integrand function to be real, the quadratic polynomial must be positive, therefore

a ) if a > 0 and y₀ is not 0 then

see the solutions of the integrals in previous examples. By using above substitutions

Note that the sign of vertical translation y₀ affects the solution, i.e., changes the sign of its second term.

b ) if a < 0 and y₀ > 0 then

see the solution of the integral in the previous example. After applying the substitutions

Recursion formulas and use of integration by parts formula

The recursion formula applied to integrals of the form

Solution:

Use the above recursion formula to calculate I₂ and I₃

Example: Evaluate

Solution:

Example: Evaluate

Solution:

The recursion formula applied to integrals of the form

Solution:

The recursion formula applied to integrals of the form

Solution:

where, the integral on the right side of the equation is already shown.

The recursion formula applied to integrals of the form

By using the substitution x - x₀ = t, this integral leads to already known type.

Solution:

Use the above solution to evaluate the following example.

Example: Evaluate

Solution

The recursion formula applied to integrals of the form

Solution

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