How this calculator works
Factor out a, add and subtract the square of half the new linear coefficient, then shift x to u. The sign of the remaining constant selects an arctangent, reciprocal or logarithmic antiderivative. The bounds may be reversed. Any real zero of the denominator in the closed interval is rejected, even if a formal subtraction of antiderivatives would look finite. This tool handles the real reciprocal-quadratic family, not arbitrary numerators or complex integration.
Formula
u = x + b/(2a); q = c/a - [b/(2a)]²; ∫dx/(ax²+bx+c) = (1/a)∫du/(u²+q). For q>0 use atan(u/√q)/(a√q); for q=0 use -1/(au); for q<0 use ln|(u-√(-q))/(u+√(-q))|/(2a√(-q)).
Worked example
For 1/(x²+4x+5), the denominator is (x+2)²+1. With u=x+2, the antiderivative is atan(x+2)+C. Integrating from -2 to -1 gives atan(1)-atan(0)=π/4.
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Frequently asked questions
Why is a required to be non-zero?
With a=0 the denominator is linear or constant. It needs a different initial reduction.
Can the completed-square constant be negative?
Yes. That branch has real poles and a logarithmic antiderivative on each interval between poles.
Does the tool compute a Cauchy principal value?
No. It rejects intervals containing a denominator zero. Cancellation across a pole is not an ordinary definite integral.