Primitive of Cosecant Function/Tangent Form

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Theorem

$\ds \int \csc x \rd x = \ln \size {\tan \frac x 2} + C$

where $\tan \dfrac x 2 \ne 0$.


Proof 1

\(\ds \int \csc x \rd x\) \(=\) \(\ds -\ln \size {\csc x + \cot x} + C\) Primitive of $\csc x$: Cosecant plus Cotangent Form
\(\ds \) \(=\) \(\ds \ln \size {\frac 1 {\csc x + \cot x} } + C\) Logarithm of Reciprocal
\(\ds \) \(=\) \(\ds \ln \size {\frac 1 {\frac 1 {\sin x} + \frac {\cos x} {\sin x} } } + C\) Definition of Cosecant and Definition of Cotangent
\(\ds \) \(=\) \(\ds \ln \size {\frac {\sin x} {1 + \cos x} } + C\)
\(\ds \) \(=\) \(\ds \ln \size {\tan \frac x 2} + C\) Half Angle Formula for Tangent: Corollary $1$

$\blacksquare$


Proof 2

\(\ds \int \csc x \rd x\) \(=\) \(\ds \int \frac 1 {\sin x} \rd x\) Cosecant is Reciprocal of Sine


We make the Weierstrass Substitution:

\(\ds u\) \(=\) \(\ds \tan \frac x 2\)
\(\ds \leadsto \ \ \) \(\ds \sin x\) \(=\) \(\ds \frac {2 u} {u^2 + 1}\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d u}\) \(=\) \(\ds \frac 2 {u^2 + 1}\)
\(\ds \leadsto \ \ \) \(\ds \int \frac 1 {\sin x} \rd x\) \(=\) \(\ds \int \frac {u^2 + 1} {2 u} \frac 2 {u^2 + 1} \rd u\)
\(\ds \) \(=\) \(\ds \int \frac 1 u \rd u\)
\(\ds \) \(=\) \(\ds \ln \size u + C\) Primitive of Reciprocal
\(\ds \) \(=\) \(\ds \ln \size {\tan \frac x 2} + C\) substituting back for $u$

$\blacksquare$


Sources

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