<p>Reverse the standard derivative formulas. For tangent and cotangent, rewrite in sine and cosine and apply the logarithmic reverse-chain-rule form.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Integrand</strong></th><th style="text-align: center; --cell-justify: center"><strong>Antiderivative</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(\sin x\)</td><td style="text-align: center; --cell-justify: center">\(-\cos x+C\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\cos x\)</td><td style="text-align: center; --cell-justify: center">\(\sin x+C\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\sec^2x\)</td><td style="text-align: center; --cell-justify: center">\(\tan x+C\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\operatorname{cosec}^2x\)</td><td style="text-align: center; --cell-justify: center">\(-\cot x+C\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\sec x\tan x\)</td><td style="text-align: center; --cell-justify: center">\(\sec x+C\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\operatorname{cosec}\,x\cot x\)</td><td style="text-align: center; --cell-justify: center">\(-\operatorname{cosec}\,x+C\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\tan x\)</td><td style="text-align: center; --cell-justify: center">\(-\ln|\cos x|+C\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(\cot x\)</td><td style="text-align: center; --cell-justify: center">\(\ln|\sin x|+C\)</td></tr></tbody></table></div></div>
<p>For an angle \(ax+b\), divide the corresponding antiderivative by \(a\), where&nbsp;\(a\ne0\). Work on an interval without a zero denominator. The derived secant and cosecant logarithmic forms are developed in the earlier logarithmic-trigonometric notes.</p>

Reverse the standard derivative formulas. For tangent and cotangent, rewrite in sine and cosine and apply the logarithmic reverse-chain-rule form.

For an angle \(ax+b\), divide the corresponding antiderivative by \(a\), where \(a\ne0\). Work on an interval without a zero denominator. The derived secant and cosecant logarithmic forms are developed in the earlier logarithmic-trigonometric notes.

Trigonometric Antiderivatives

Trigonometric Antiderivatives

Video Solution:

Integrand	Antiderivative
\(\sin x\)	\(-\cos x+C\)
\(\cos x\)	\(\sin x+C\)
\(\sec^2x\)	\(\tan x+C\)
\(\operatorname{cosec}^2x\)	\(-\cot x+C\)
\(\sec x\tan x\)	\(\sec x+C\)
\(\operatorname{cosec}\,x\cot x\)	\(-\operatorname{cosec}\,x+C\)
\(\tan x\)	\(-\ln\|\cos x\|+C\)
\(\cot x\)	\(\ln\|\sin x\|+C\)