Arctangent Terms From Constant Quadratic Numerators

Introduction

A rational function with an irreducible quadratic denominator may have a constant numerator. That constant-numerator term has a characteristic antiderivative: an inverse tangent. The formula is simple, but it contains a scale factor that is easy to overlook.

For instance, the denominator \(x^2+9\) does not lead to just a constant times \(\arctan x\). The number \(9\) determines both the argument of the inverse tangent and the coefficient in front of it. We will derive the general rule, apply it to exact examples, and check each answer by differentiation.

The Standard Form

In this tutorial, the standard quadratic denominator is \(x^2+a^2\), where \(a\) is a positive real number. It is positive for every real \(x\), so the rational function \(C/(x^2+a^2)\) is defined on the entire real line when \(C\) is constant.

The corresponding integration rule is

where \(K\) is an arbitrary constant of integration. The parameter \(a\) is the positive square root of the constant term in the denominator: if the denominator is \(x^2+b\) with \(b>0\), then \(a=\sqrt b\). Writing the denominator this way makes the scale factor explicit.

The constant numerator matters. A numerator proportional to \(x\) is matched to the derivative of the denominator and gives a logarithmic term. A constant numerator over \(x^2+a^2\), by contrast, produces an inverse tangent. In partial fractions, a constant numerator over an irreducible quadratic is therefore a signal to use this rule.

Why the Scale Factor Appears

We can derive the rule by differentiating the proposed inverse-tangent expression. The derivative of \(\arctan u\) is \(1/(1+u^2)\), and the chain rule accounts for the factor from its argument. For \(a>0\),

Thus the derivative of \(\frac{C}{a}\arctan(x/a)\) is exactly the desired integrand. This verifies the formula and shows why the coefficient \(1/a\) is required: the derivative of the inverse tangent itself contributes \(a\) in the numerator.

The same factor can be seen by a substitution. Set \(u=x/a\), so \(x=au\) and \(dx=a\,du\). The denominator becomes \(a^2(u^2+1)\), giving

Substituting \(u=x/a\) gives the stated formula. Both derivations reveal the same scale factor. The condition \(a>0\) fixes the scale as a positive number; since \(a\) comes from a square root, this is the natural convention.

Worked Examples

A Reliable Way to Apply the Rule

When you encounter a constant-numerator term over a standard quadratic, keep the parameter and the coefficient distinct. First identify \(a\) from the square \(a^2\), then divide the numerator constant \(C\) by \(a\), and finally place \(x/a\) inside the inverse tangent. This order helps prevent the two scale factors from being confused.

This rule applies directly when the denominator is exactly \(x^2+a^2\) with \(a>0\). A denominator that is shifted or has a coefficient other than \(1\) on \(x^2\) does not match this standard form as written; do not silently use the formula without accounting for those changes. The special case \(a=0\) is also excluded: then the denominator is \(x^2\), not a positive quadratic of this form.

Common Pitfall: Omitting the Scale Factor

A frequent error is to notice the inverse tangent but write an unscaled argument and coefficient. For example, one might incorrectly claim

Differentiating the proposed answer exposes the error:

which is not \(6/(x^2+9)\). The denominator is wrong, so the proposed expression cannot be an antiderivative. The correct parameter is \(a=3\). Both places where it matters must be included: divide the coefficient by \(3\) and use \(x/3\) as the argument.

Checking by differentiation is especially useful here because it tests both scale factors at once. If either the outside coefficient or the argument is incorrect, the derivative will not return the original rational term.

Why This Form Matters

Irreducible quadratic factors occur in partial fraction decompositions, where the numerator over such a factor is linear. The numerator can be separated into a part proportional to the derivative of the quadratic and a constant part. The earlier material on logarithmic terms treats the derivative-matched part; this rule handles the constant part when the denominator has the standard form \(x^2+a^2\).

Recognizing which numerator structure is present saves unnecessary algebra. The logarithmic and inverse-tangent patterns are different because the numerator matches different features of the denominator: its derivative in one case, and a constant over the sum of squares in the other. Once the term is in the required standard form, the antiderivative rule is direct.