Solve integrals of the form ∫ dx/(ax² + bx + c) by completing the square. Shows the completed square form, integral type (arctan or partial fractions), step-by-step solution, and formula references.
<p>The <strong>Integration by Completing the Square Calculator</strong> solves integrals of the form ∫ dx / (ax² + bx + c) using the completing-the-square technique, one of the most important methods in calculus for handling quadratic denominators.</p> <p>When the denominator is a quadratic that cannot be factored easily, completing the square rewrites ax² + bx + c into the form a[(x + p)² ± q²]. Depending on the sign of the discriminant, the integral evaluates to an arctangent function (when the quadratic has no real roots) or decomposes via partial fractions (when the quadratic factors over the reals).</p> <p>This calculator walks you through each step: extracting the leading coefficient, completing the square, identifying the resulting integral form, and presenting the final antiderivative. It handles all cases — positive definite quadratics leading to arctan, negative discriminant cases, and factorable quadratics using partial fractions with logarithms.</p> <p>Whether you're a calculus student preparing for an exam, a physicist evaluating probability integrals, or an engineer solving transfer-function problems, this tool provides the step-by-step algebraic work that textbooks often skip, along with a reference table of common integral forms and preset coefficient sets for practice.</p>
This calculator is useful when a quadratic denominator does not factor cleanly and you need to decide which integration path actually applies. Instead of guessing whether the result should involve arctangent, logarithms, or a repeated-factor power rule, the tool completes the square, checks the discriminant, and shows the matching antiderivative form.
That makes it valuable for calculus homework, exam review, and self-checking symbolic work. It does not just return an answer; it exposes the intermediate algebra so you can see how the completed-square form connects to the standard integral formulas used in textbooks.
Completing the square: ax² + bx + c = a[(x + b/2a)² + (4ac − b²)/(4a²)] If 4ac − b² > 0: ∫dx/(ax²+bx+c) = (2/√(4ac−b²)) arctan((2ax+b)/√(4ac−b²)) + C If 4ac − b² < 0: uses partial fractions with logarithms
Result: 1/2 arctan((x + 2) / 2) + C
Enter aStr = 1, bStr = 4, and cStr = 8. The calculator rewrites x² + 4x + 8 as (x + 2)² + 4, so the integral becomes ∫ dx / [(x + 2)² + 2²]. Because the discriminant is negative, the quadratic is irreducible over the reals and the antiderivative is the arctan form: 1/2 arctan((x + 2) / 2) + C.
An integral like $int rac{dx}{ax^2 + bx + c}$ is hard to recognize in its raw form because the denominator hides the standard pattern. Completing the square rewrites the quadratic into a shifted square plus or minus a constant, which reveals whether the denominator behaves like $(x+p)^2 + k^2$, $(x+p)^2 - k^2$, or a perfect square. Once that form appears, the antiderivative is no longer mysterious.
This is the same algebraic move used to derive the quadratic formula, but in calculus the goal is different. Instead of solving for roots, you are transforming the denominator into something that matches a memorized integration formula.
The discriminant $Delta = b^2 - 4ac$ tells you which case you are in. If $Delta < 0$, the quadratic has no real roots and the completed-square form leads to an arctangent result. If $Delta = 0$, the denominator becomes a repeated linear factor and the antiderivative reduces to a power-rule style expression. If $Delta > 0$, the quadratic factors over the reals and partial fractions produce logarithms.
That classification is one of the main teaching points of this calculator. It links a familiar algebra concept to the calculus method that follows from it, so you can predict the answer form before carrying out the full integration.
Completing the square is especially useful when factoring is awkward or impossible by inspection. On homework, it helps you write cleaner solutions because every transformation has a clear reason. On exams, it helps you avoid wasting time trying random factorizations that do not exist.
A good workflow is to factor out the leading coefficient first when $a eq 1$, complete the square inside the brackets, and then compare the result to the standard reference forms. That makes it easier to spot sign mistakes and to justify why the final antiderivative should involve arctangent, logarithms, or a reciprocal linear term.
Whenever you have an integral with a quadratic expression in the denominator that doesn't factor neatly, completing the square lets you rewrite it into a standard form with a known antiderivative. Understanding this concept helps you apply the calculator correctly and interpret the results with confidence.
If the quadratic has no real roots (Δ < 0), you get an arctan integral. If it has real roots (Δ > 0), you factor and use partial fractions. If Δ = 0, you get a power-rule integral.
Yes. Factoring out a negative from the quadratic changes the sign, and the calculator adjusts the resulting integral type accordingly.
If a = 0, the expression is linear (bx + c), not quadratic. The integral is simply (1/b) ln|bx + c| + C. The calculator will indicate this special case.
Δ = b² − 4ac. Negative → arctan (irreducible quadratic), zero → perfect square (power rule), positive → partial fractions (logarithms).
They are closely related. The quadratic formula is derived by completing the square on ax² + bx + c = 0. For integration, you complete the square in the denominator rather than solving = 0.