Primitive of Sine Function/Corollary

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Corollary to Primitive of Sine Function

$\displaystyle \int \sin a x \rd x = - \frac {\cos a x} a + C$

where $C$ is an arbitrary constant.


Proof

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

$\blacksquare$


Also see


Sources