Integration By Completing The Square Calculator - Rewrite, Substitute, Integrate

An integration by completing the square calculator evaluates the indefinite integral of 1/(ax^2 + bx + c) by rewriting the quadratic denominator, applying a u-substitution, and returning the matching arctan, logarithm, or rational antiderivative. Type the three coefficients a, b, c, and the page reports the completed-square form, the u-substitution, the branch, and the closed-form answer in x plus C.

• • Calculus 2 Homework: Solve textbook problems that ask for the antiderivative of 1/(x^2 + 1), 1/(x^2 + 2x + 2), or 1/(2x^2 + 4x + 4).
• • Definite Integral Setup: Use a closed-form antiderivative of 1/(ax^2 + bx + c) to plug into a definite integral, especially in physics or probability problems.
• • Verifying Partial-Fraction Steps: Cross-check an answer produced by hand or by a partial-fraction-decomposition calculator, especially the arctan branch when the denominator has no real roots.

The same completing-the-square rewrite that turns ax^2 + bx + c into a(x - h)^2 + k also drives this page's integral, and the standalone completing the square calculator applies the same rewrite to vertex-form problems once a u-substitution is added on top.

The page reads the three coefficients a, b, c, completes the square in the denominator, applies the matching u-substitution, and reduces the integral to a standard arctan, logarithm, or simple rational antiderivative.

• a: Leading coefficient. Must be non-zero; if a = 0 the integrand is no longer a proper quadratic.
• b: Linear coefficient. Sets the horizontal shift of the completed square at x = -b/(2a).
• c: Constant term. Together with a and b it sets the discriminant sign.
• D = b^2 - 4ac: Negative D means no real roots and the answer is arctan. Zero D means a perfect square and the answer is a simple rational. Positive D means two real roots and the answer is a logarithm.

For a = 1, b = 0, c = 1 the integral of 1/(x^2 + 1) dx returns arctan(x) + C. For a = 1, b = 2, c = 5 the inner arctan argument becomes (x + 1) / 2, producing 0.5 * arctan((x + 1) / 2) + C, which shows how the scale inside the arctan tracks the completed-square constant. For two-real-root inputs, such as a = 1, b = -3, c = 2, the page returns the logarithm branch with the two roots in the inner fraction, while a = 2, b = -6, c = 4 returns 0.5 * ln |(x - 2) / (x - 1)| + C.

According to Wikipedia, completing the square rewrites ax^2 + bx + c as a*(x - h)^2 + k with h = -b/(2a) and k = c - b^2/(4a), and the same rewrite lets you evaluate the integral of 1/(ax^2 + bx + c) using the basic arctan and logarithm integrals.

The discriminant sign that drives the branch selection here is the same discriminant that the quadratic formula calculator uses to count the real roots of ax^2 + bx + c, so the two tools agree on every input that crosses the D = 0 boundary.

Four ideas make the technique work, and recognizing each one keeps the integral honest when the coefficients get unusual.

These four ideas are the only machinery behind the technique, and the page builds each one into a separate line in the result card.

The arctan branch lands on the standard arctan integral, and the companion inverse tangent calculator evaluates arctan(x) at a specific x so you can sanity-check the closed-form coefficient by plugging a sample value.

Follow four short steps to turn the three coefficients of the quadratic denominator into a closed-form antiderivative.

1. 1 Enter the Leading Coefficient a: Type the coefficient in front of x^2. The default 1 matches the standard arctan integral of 1/(x^2 + 1). The value must be non-zero; the page returns a clear error when a = 0.
2. 2 Enter the Linear Coefficient b: Type the coefficient in front of x. The default 0 keeps the parabola centered on the y-axis, while a non-zero b shifts the vertex of the completed square.
3. 3 Enter the Constant c: Type the constant term. Together with a and b, c sets the discriminant sign.
4. 4 Read the Four Result Lines: The result branch line names the family. The completed-square line shows the rewritten denominator. The u-substitution line shows u and du. The antiderivative line shows the final closed form plus C.

For int 1/(2x^2 + 4x + 4) dx, type a = 2, b = 4, c = 4 and the page returns the arctan branch with completed-square form 2 * (x + 1)^2 + 2 and the final antiderivative 0.5 * arctan(x + 1) + C.

When the denominator is a product of quadratics instead of a single trinomial, the partial fraction decomposition calculator produces the per-factor pieces first and the same completing-the-square rewrite then reappears inside each quadratic partial fraction.

A dedicated integration by completing the square calculator removes the small mistakes that make hand-evaluated integrals of 1/(ax^2 + bx + c) go wrong.

• • Three Result Branches in One Place: The same page returns the arctan (D < 0), logarithm (D > 0), and simple rational (D = 0) branches, so there is no need to switch tools when the discriminant sign changes.
• • Completed-Square and u-Substitution Lines: The page reports the rewrite and the substitution on two separate lines, so the steps that turn a difficult integral into a basic one stay visible.
• • Discriminant Sign Made Explicit: The result branch line states which family the discriminant selected, so it is never ambiguous whether the answer should be an arctan, a logarithm, or a simple rational.
• • Clean Handling of the a = 0 Edge Case: When the leading coefficient is zero the technique does not apply, and the page surfaces a clear error message instead of returning NaN or Infinity.
• • Hand-off to Peer Calculators: The completed-square line lines up with the completing-the-square calculator, the discriminant sign with the quadratic formula calculator, and the arctan branch with the inverse-tangent calculator on the same site.

These benefits matter most in second-semester calculus, where one wrong sign in the discriminant can change the final answer.

When the numerator has higher degree than the quadratic denominator, the polynomial division calculator simplifies the problem to a polynomial plus a proper rational function so the technique on this page can take over.

Three measurable factors control which branch the integral takes, and two practical limits apply to any hand-evaluated or tool-evaluated integral of 1/(ax^2 + bx + c).

• • The closed-form antiderivative is exact only when the denominator is a single quadratic. For products of quadratics or a rational function with a non-trivial numerator, run a partial-fraction-decomposition-calculator pass first; this page will not produce the correct answer alone.
• • Inputs are bounded between -100 and 100 to keep the closed form readable. Values outside that range can overflow the arctan or logarithm formulas, so the page surfaces a clear message.

These factors and limits are the only practical concerns for a single-quadratic denominator. For products of linear or quadratic factors, the partial-fraction-decomposition-calculator handles the next step up.

According to Paul's Online Math Notes, completing the square on a non-monic quadratic like 2x^2 - 3x + 2 turns the integral into the canonical arctan form, with the general antiderivative (2/sqrt(4ac - b^2)) * arctan((2*a*x + b)/sqrt(4ac - b^2)) + C when 4ac - b^2 > 0.

According to Omni Calculator, the same rewrite also drives the integral of 1/(ax^2 + bx + c) dx through the u = sqrt(|a|)*(x + b/(2a)) substitution, reducing the integrand to the standard arctan integral of 1/(1 + u^2).

The two real roots that show up in the logarithm branch of this calculator are the same roots the quadratic equation solver returns for ax^2 + bx + c = 0, so you can read them off the equation solver and paste them into the inner fraction on the result card.