Antiderivatives Of Tangent And Cotangent

Why Tangent and Cotangent Lead to Logarithms

Tangent and cotangent are quotients of sine and cosine. Their antiderivatives are logarithmic because the derivative of the natural logarithm of a nonzero function \(u(x)\) is \(u'(x)/u(x)\). In these trigonometric quotients, the numerator that appears after differentiating the logarithm matches the other trigonometric function in the quotient.

The signs require care. The cosine derivative is negative sine, while the sine derivative is cosine. As a result, the tangent antiderivative has a minus sign and the cotangent antiderivative does not.

All angles in the formulas below are measured in radians. Each formula applies on an interval where the trigonometric function being integrated is defined.

Deriving the Tangent Formula

Recall that \(\tan x=\sin x/\cos x\), so tangent is defined wherever \(\cos x\ne 0\). On any interval with this property, \(\cos x\) never equals zero. The logarithm of its absolute value is therefore defined throughout the interval.

For any differentiable function \(u\) that is nonzero, the chain rule gives \(\frac{d}{dx}\ln|u(x)|=u'(x)/u(x)\). Apply this fact to \(u(x)=\cos x\). Since \(\frac{d}{dx}\cos x=-\sin x\), we obtain:

Thus the negative of this logarithm differentiates to tangent:

Therefore, on any interval where \(\cos x\ne 0\), the family of antiderivatives is:

Because \(\sec x=1/\cos x\) wherever cosine is nonzero, the same answer can also be written as \(\ln|\sec x|+C\). The two logarithmic forms differ only in how the reciprocal is expressed.

Deriving the Cotangent Formula

Cotangent is \(\cot x=\cos x/\sin x\), so it is defined wherever \(\sin x\ne 0\). On an interval where sine is nonzero, apply the logarithmic derivative rule to \(u(x)=\sin x\). Since \(\frac{d}{dx}\sin x=\cos x\), we have:

It follows that on any interval where \(\sin x\ne 0\):

The two derivations have the same structure: take the logarithm of the denominator of the quotient, differentiate it, and compare the resulting quotient with the trigonometric function. The derivative of cosine introduces a minus sign, whereas the derivative of sine does not.

Worked Examples

Why the Absolute Values Matter

The expressions \(\ln(\cos x)\) and \(\ln(\sin x)\) are real-valued only when their arguments are positive. But tangent and cotangent are defined on intervals where their denominators can be negative as well as positive. The expressions \(\ln|\cos x|\) and \(\ln|\sin x|\) remain real whenever the corresponding denominator is nonzero.

The absolute value does not change the derivative formula on intervals where \(u\ne 0\): \(\frac{d}{dx}\ln|u(x)|=u'(x)/u(x)\). Thus it preserves the desired quotient while allowing the formula to apply on both positive and negative portions of the domain.

For example, replacing \(\ln|\cos x|\) with \(\ln(\cos x)\) unnecessarily restricts the expression to points where \(\cos x>0\). It cannot serve as a real-valued formula on intervals where cosine is negative, even though tangent is defined there.

A Common Sign Mistake

A frequent error is to write \(\int \tan x\,dx=\ln|\cos x|+C\), omitting the negative sign. Differentiate that proposed answer to test it:

This gives the negative of the required integrand, so the proposed antiderivative is incorrect. The correction is to negate the logarithm:

Checking by differentiation is especially useful here because the sine and cosine derivatives have different signs. A second common mistake is to leave out the absolute value. Doing so can make the logarithm undefined on part of the interval where the original trigonometric function is defined.

Summary

The logarithmic antiderivatives follow directly from the logarithmic derivative rule and the quotient definitions of tangent and cotangent. The sign in the tangent formula comes from \(\frac{d}{dx}\cos x=-\sin x\); the cotangent formula uses \(\frac{d}{dx}\sin x=\cos x\) and has no extra minus sign.