Primitive of Sine of a x + b/Proof 1

Corollary to Primitive of Sine Function

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

Proof

\(\ds \int \sin x \rd x\)

\(=\)

\(\ds -\cos x + C\)

Primitive of $\sin x$

\(\ds \leadsto \ \ \)

\(\ds \int \map \sin {a x + b} \rd x\)

\(=\)

\(\ds \frac 1 a \paren {-\map \cos {a x + b} } + C\)

Primitive of Function of $a x + b$

\(\ds \)

\(=\)

\(\ds -\frac {\map \cos {a x + b} } a + C\)

simplifying

$\blacksquare$