Integrals of Secant-Squared and Cosecant-Squared

From Trigonometric Derivatives to Antiderivatives

The derivative rules for tangent and cotangent immediately provide two useful antiderivative rules. The key is to recognize both the trigonometric expression and, when its argument is not simply \(x\), the constant factor contributed by the chain rule.

These rules apply on intervals where the expressions are defined. In particular, \(\sec^2 x\) is undefined when \(\cos x=0\), and \(\csc^2 x\) is undefined when \(\sin x=0\). An antiderivative must be considered on an interval that does not cross one of those undefined points.

The Basic Antiderivative Rules

The derivative of tangent is \(\sec^2 x\), and the derivative of cotangent is \(-\csc^2 x\). Reversing these derivative rules gives the corresponding antiderivatives:

The negative sign in the second formula is essential. Since \(\frac{d}{dx}(\cot x)=-\csc^2 x\), the antiderivative of positive \(\csc^2 x\) is \(-\cot x\), not \(\cot x\).

Why Linear Arguments Require a Factor

Suppose the argument is \(kx+b\), where \(k\) and \(b\) are constants. Differentiating \(\tan(kx+b)\) produces not just \(\sec^2(kx+b)\), but also the derivative \(k\) of the inner expression. That factor must be accounted for in the antiderivative.

When \(k\ne 0\), multiplying \(\tan(kx+b)\) by \(1/k\) cancels that extra factor. The same reasoning applies to cotangent, with its negative derivative.

For the secant-squared formula, the interval must satisfy \(\cos(kx+b)\ne 0\). For the cosecant-squared formula, it must satisfy \(\sin(kx+b)\ne 0\). The constant \(C\) can be chosen independently on each interval of definition.

Deriving and Verifying the Rules

The formulas follow from the chain rule and the derivatives of tangent and cotangent. Those derivative identities can also be seen from the quotient rule. Where \(\cos x\ne 0\), tangent is \(\sin x/\cos x\), so:

Where \(\sin x\ne 0\), cotangent is \(\cos x/\sin x\), and the quotient rule gives:

Now differentiate each proposed antiderivative for a linear argument. For secant-squared:

For cosecant-squared:

Each derivative reproduces its integrand. This verifies both formulas and explains why the coefficient is \(1/k\) and why the cosecant-squared antiderivative has a negative sign.

Constant Factors Outside the Trigonometric Function

An integrand may include a constant multiplier as well as a linear argument. In that case, keep the outside multiplier and the argument’s derivative visible. The result can then be checked by differentiating it.

When the Argument Has No Variable Part

The linear-argument formulas above require \(k\ne 0\), because they divide by \(k\). If \(k=0\), the argument \(kx+b\) is simply the constant \(b\). Provided the trigonometric expression is defined, the integrand is a constant function:

These follow because the derivative of a constant multiple of \(x\) is that constant. If the relevant denominator is zero, the integrand is undefined and these expressions do not apply.

Common Errors and Why the Rules Matter

A frequent error is to forget the factor from differentiating the argument. For example, an incorrect attempt is:

Differentiate the proposed answer:

which is three times the integrand, not the integrand itself. The correction is to include the reciprocal factor:

Another error is to omit the negative sign for cosecant-squared. Since differentiating \(\cot x\) gives \(-\csc^2 x\), writing \(\cot x+C\) as an antiderivative of \(\csc^2 x\) has the wrong sign. Checking by differentiation catches both kinds of error quickly.

These antiderivatives are useful because they connect familiar trigonometric derivatives to integration patterns. Recognizing the pattern, tracking the derivative of the argument, and verifying the answer by differentiation provide a reliable method without relying on guesswork.