Primitive of Cosine of a x + b/Proof 3

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Corollary to Primitive of Cosine Function

$\ds \int \map \cos {a x + b} \rd x = \frac {\map \sin {a x + b} } a + C$


\(\ds \map {\dfrac \d {\d x} } {\frac {\map \sin {a x + b} } a}\) \(=\) \(\ds \dfrac 1 a \map \cos {a x + b} \map {\dfrac \d {\d x} } {a x + b}\) Derivative of Sine Function, Chain Rule for Derivatives
\(\ds \) \(=\) \(\ds \dfrac 1 a \cdot a \map \cos {a x + b}\) Power Rule for Derivatives
\(\ds \) \(=\) \(\ds \cos {a x + b}\) simplifying

The result follows by definition of primitive.