Checking Tangent, Secant, Cotangent, And Cosecant Antiderivatives

Why Check an Antiderivative?

An antiderivative is a function whose derivative equals a given integrand. Trigonometric integrals often produce logarithms, and a sign or constant-factor error can be difficult to spot by looking at the final expression alone. Differentiation provides a direct check: a proposed answer is correct on an interval if its derivative equals the integrand at every point of that interval.

This tutorial checks common antiderivatives involving tangent, secant, cotangent, and cosecant. The main points are to keep the chain-rule factor, handle the signs of cotangent and cosecant carefully, and distinguish expressions that look different but differ only by an equivalent logarithmic form.

The Derivatives That Control the Checks

The standard derivative identities needed here are

For a composite angle \(u=g(x)\), the chain rule supplies a factor \(u'=g'(x)\). The logarithmic derivative rule, \(\frac{d}{dx}\ln|h(x)|=\frac{h'(x)}{h(x)}\) when \(h(x)\ne0\), then turns the proposed logarithms into ratios that can be simplified.

Checking the Four Basic Forms

Each formula below is valid on any interval on which the trigonometric expressions involved are defined. In particular, tangent and secant require \(\cos x\ne0\), while cotangent and cosecant require \(\sin x\ne0\).

Indeed, \(\frac{d}{dx}[-\ln|\cos x|]=-\frac{-\sin x}{\cos x}=\tan x\). Also, \(\ln|\sec x|=-\ln|\cos x|\), because \(\sec x=1/\cos x\).

Differentiating the logarithm gives \(\frac{\sec x\tan x+\sec^2 x}{\sec x+\tan x}=\sec x\). The numerator factors as \(\sec x(\tan x+\sec x)\), so the denominator cancels.

The derivative is \(\frac{\cos x}{\sin x}=\cot x\). This formula has a positive sign: the derivative of \(\ln|\sin x|\) is positive \(\cot x\), even though the derivative of \(\cot x\) itself is negative.

To check it, differentiate the logarithm. The numerator of the resulting ratio is \(-\csc x\cot x+\csc^2 x=\csc x(\csc x-\cot x)\), so division by \(\csc x-\cot x\) leaves \(\csc x\).

Why the Secant and Cosecant Logarithms Work

The secant formula can be derived by multiplying the integrand by a carefully chosen form of 1. This creates the derivative of the logarithm’s argument in the numerator.

The cosecant formula follows the same principle, with the signs determined by the derivatives of cosecant and cotangent:

Consequently, on an interval where \(\sin x\ne0\),

This derivation also explains why changing the sign inside the cosecant logarithm without changing anything else is not harmless. In fact, the other common expression has the opposite logarithmic derivative:

Equivalent Logarithmic Forms

Two logarithmic antiderivatives can look different while having the same derivative. The identities below show exactly how the common forms are related. Their absolute values are important because the expressions inside the logarithms need not be positive.

On intervals where \(\cos x\ne0\), neither factor is zero. Taking logarithms of absolute values gives

Similarly, where \(\sin x\ne0\),

so

The negative sign in the last identity is essential. It is why \(\ln|\csc x+\cot x|\) alone is an antiderivative of \(-\csc x\), not of \(\csc x\).

Worked Examples

A Common Sign and Chain-Rule Mistake

A frequent error is to copy the cotangent formula with a negative sign, or to write a formula for an inner angle without its derivative factor. For example, the incorrect claim

fails when checked: with \(u=3x\), its derivative is

not \(\cot(3x)\). The correction is to use a positive sign and divide by the inner derivative:

Checking by differentiation catches both errors at once: the sign and the missing chain-rule factor.

How to Check a Proposed Formula Reliably

Use the following routine whenever a tangent, secant, cotangent, or cosecant antiderivative is in doubt:

The domain check is part of the verification, not an optional detail. An antiderivative is considered on an interval where the original integrand is defined; the derivative calculation must hold throughout that interval. For example, a formula involving \(\cot x\) cannot apply across a point where \(\sin x=0\).