Primitive of Cosecant Function/Tangent Form

Theorem

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

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

Proof 1

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Since:

$\tan \dfrac x 2 \ne 0$

It follows that $\forall n \in \Z$:

$ 2\pi n < x < 2\pi n + \pi$

So:

$\csc x + \cot x \ne 0$

And:

$\dfrac {1 + \cos x} {\sin x} \ne 0$

Hence:

$\ds \int \csc x \rd x$

$=$

$\ds -\ln \size {\csc x + \cot x} + C$

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

1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Integration
1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Integration
1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xix)}$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.16$
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1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 16$: Indefinite Integrals: General Rules of Integration: $16.16.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $7$: Integrals
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $8$: Integrals