The following results can be derived from the logarithmic reverse chain rule. They are useful consequences, not extra formula-booklet entries to assume.
Tangent
\(\begin{aligned}\int\tan x\,\mathrm{d}x&=\int\dfrac{\sin x}{\cos x}\,\mathrm{d}x\\&=-\ln|\cos x|+C\end{aligned}\)
Cotangent
\(\begin{aligned}\int\cot x\,\mathrm{d}x&=\int\dfrac{\cos x}{\sin x}\,\mathrm{d}x\\&=\ln|\sin x|+C\end{aligned}\)
Secant
\(\begin{aligned}\int\sec x\,\mathrm{d}x&=\int\dfrac{\sec^2x+\sec x\tan x}{\sec x+\tan x}\,\mathrm{d}x\\&=\ln|\sec x+\tan x|+C\end{aligned}\)
Cosecant
\(\begin{aligned}\int\operatorname{cosec}\,x\,\mathrm{d}x&=\int\dfrac{\operatorname{cosec}^2x-\operatorname{cosec}\,x\cot x}{\operatorname{cosec}\,x-\cot x}\,\mathrm{d}x\\&=\ln|\operatorname{cosec}\,x-\cot x|+C\end{aligned}\)
Differentiate each logarithm to check the sign. Apply the linear scale factor or a supplied substitution when the angle is more complicated.

The following results can be derived from the logarithmic reverse chain rule. They are useful consequences, not extra formula-booklet entries to assume.

\(\begin{aligned}\int\tan x\,\mathrm{d}x&=\int\dfrac{\sin x}{\cos x}\,\mathrm{d}x\\&=-\ln|\cos x|+C\end{aligned}\)

\(\begin{aligned}\int\cot x\,\mathrm{d}x&=\int\dfrac{\cos x}{\sin x}\,\mathrm{d}x\\&=\ln|\sin x|+C\end{aligned}\)

\(\begin{aligned}\int\sec x\,\mathrm{d}x&=\int\dfrac{\sec^2x+\sec x\tan x}{\sec x+\tan x}\,\mathrm{d}x\\&=\ln|\sec x+\tan x|+C\end{aligned}\)

\(\begin{aligned}\int\operatorname{cosec}\,x\,\mathrm{d}x&=\int\dfrac{\operatorname{cosec}^2x-\operatorname{cosec}\,x\cot x}{\operatorname{cosec}\,x-\cot x}\,\mathrm{d}x\\&=\ln|\operatorname{cosec}\,x-\cot x|+C\end{aligned}\)

Differentiate each logarithm to check the sign. Apply the linear scale factor or a supplied substitution when the angle is more complicated.

Deriving Logarithmic Trigonometric Forms

Deriving Logarithmic Trigonometric Forms