Primitive of Square of Sine of a x

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Theorem

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


Proof

\(\displaystyle \int \sin^2 x \rd x\) \(=\) \(\displaystyle \frac x 2 - \frac {\sin 2 x} 4 + C\) Primitive of $\sin^2 x$
\(\displaystyle \leadsto \ \ \) \(\displaystyle \int \sin^2 a x \rd x\) \(=\) \(\displaystyle \frac 1 a \paren {\frac {a x} 2 - \frac {\sin 2 a x} 4} + C\) Primitive of Function of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \frac x 2 - \frac {\sin 2 a x} {4 a} + C\) simplifying

$\blacksquare$


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