Primitive of Square of Sine Function

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Theorem

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

where $C$ is an arbitrary constant.


Corollary

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

where $C$ is an arbitrary constant.


Proof

\(\ds \int \sin^2 x \rd x\) \(=\) \(\ds \int \paren {\frac {1 - \cos 2 x} 2} \rd x\) Square of Sine
\(\ds \) \(=\) \(\ds \frac 1 2 \int \d x - \frac 1 2 \int \cos 2 x \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac x 2 - \frac 1 2 \int \cos 2 x \rd x + C\) Primitive of Constant
\(\ds \) \(=\) \(\ds \frac x 2 - \frac 1 2 \paren {\frac {\sin 2 x} 2} + C\) Primitive of $\cos a x$
\(\ds \) \(=\) \(\ds \frac x 2 - \frac {\sin 2 x} 4 + C\) rearranging

$\blacksquare$


Also presented as

Some sources present this as:

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


Sources