Primitive of Square of Sine Function/Corollary

Corollary to Primitive of Square of Sine Function

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

where $C$ is an arbitrary constant.

Proof

\(\ds \int \sin^2 x \rd x\)

\(=\)

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

Primitive of Square of Sine Function

\(\ds \)

\(=\)

\(\ds \frac x 2 - \frac {2 \sin x \cos x} 4 + C\)

Double Angle Formula for Sine

\(\ds \)

\(=\)

\(\ds \frac {x - \sin x \cos x} 2 + C\)

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 14$: General Rules of Integration: $14.21$
• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $6$. Integral Calculus: Appendix: Table of Fundamental Standard Integrals