Complete the Square for Integrals
In calculus, integrating rational functions often requires completing the square in the denominator to match standard integration formulas like arctan.
Complete the Square (Integrals)
Calculus Applications
When faced with an integral of the form ∫ 1/(ax² + bx + c) dx, standard substitution usually fails. By completing the square, you can transform the denominator into the form u² + a², which allows for trigonometric substitution.
The Arctan Rule
The most common result of completing the square in an integral denominator is the arctangent function. The standard formula is ∫ 1/(x² + a²) dx = (1/a)arctan(x/a) + C.
Handling Roots
This technique is also essential when the quadratic is under a square root in the denominator: ∫ 1/√(ax² + bx + c) dx. Completing the square transforms this into an arcsin or natural log form, depending on the signs.
Complete the Square Calculators
Specialized tools for every completing the square scenario — pick the one that matches your problem.