Integrals Involving Quadratics
When an integral contains a quadratic expression that doesn't factor nicely, we Complete the Square to transform it into a form where we can use inverse trig rules.
Completing the Square
When an integral contains a quadratic expression like in the denominator (inside a root or fraction), and it doesn't factor nicely, we Complete the Square.
This transforms the expression into a form where we can use Trig Substitution or Inverse Trig rules.
The Completing the Square Formula
For any quadratic :
When :
Two Resulting Forms
Sum of Squares
→ Use formula
Difference of Squares
→ Use formula
Key Integration Formulas
Arctangent Form
Use when you have in the denominator.
Arcsine Form
Use when you have in the denominator.
Logarithm Form
Use when the numerator is (or contains) the derivative of the quadratic.
Splitting Numerators
If the numerator is , split into two integrals:
Worked Examples
Example 1: Arctangent Form
Evaluate .
The denominator doesn't factor (discriminant ).
Take half of , square it:
This is — matches form!
Let , so :
Using with :
Example 2: Arcsine Form
Evaluate .
We have a square root of a quadratic. Need to complete the square inside.
Rewrite:
Complete square inside:
This is — matches form!
Let , so :
Using with :
Example 3: Splitting Numerators
Evaluate .
Let , so and :
The numerator is (half of) the derivative of the denominator → ln form
Standard inverse tangent form → arctan
Note: so we can use the original expression.
Practice Quiz
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