Primitive of Square of Sine Function

Theorem

$\ds \int \sin^2 x \rd x = \frac x 2 - \frac {\sin 2 x} 4 + C$

where $C$ is an arbitrary constant.

$\ds \int \sin^2 x \rd x = \frac {x - \sin x \cos x} 2 + C$

$\ds \int \sin^2 x \rd x$

$=$

$\ds \int \paren {\frac {1 - \cos 2 x} 2} \rd x$

$\ds $

$=$

$\ds \frac 1 2 \int \d x - \frac 1 2 \int \cos 2 x \rd x$

$\ds $

$=$

$\ds \frac x 2 - \frac 1 2 \int \cos 2 x \rd x + C$

$\ds $

$=$

$\ds \frac x 2 - \frac 1 2 \paren {\frac {\sin 2 x} 2} + C$

$\ds $

$=$

$\ds \frac x 2 - \frac {\sin 2 x} 4 + C$

rearranging

$\blacksquare$

Some sources present this as:

$\ds \int \sin^2 x \rd x = \frac 1 2 \paren {x - \frac {\sin 2 x} 2} + C$

1960: Margaret M. Gow: A Course in Pure Mathematics ... (previous) ... (next): Chapter $10$: Integration: $10.4$. Standard integrals: Other Standard Results: $\text {(xxi)}$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.21$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Appendix: Table $2$: Integrals
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $2$: 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: $16.21.$
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