Primitive of Cosine of a x + b/Proof 3

Corollary to Primitive of Cosine Function

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

Proof

\(\ds \map {\dfrac \d {\d x} } {\frac {\map \sin {a x + b} } a}\)

\(=\)

\(\ds \dfrac 1 a \map \cos {a x + b} \map {\dfrac \d {\d x} } {a x + b}\)

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.

$\blacksquare$

Sources

• 1945: A. Geary, H.V. Lowry and H.A. Hayden: Advanced Mathematics for Technical Students, Part I ... (previous) ... (next): Chapter $\text {III}$: Integration: Three rules for integration: $\text {III}$