Primitive of Sine of a x + b/Examples/3 - x

Examples of Use of Primitive of $\map \sin {a x + b}$

$\ds \int \map \sin {3 - x} \rd x = \map \cos {3 - x} + C$

Proof

From Primitive of $\map \sin {a x + b}$:

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

\(\ds \int \map \sin {3 - x} \rd x\)

\(=\)

\(\ds -\frac {\map \cos {\paren {-1} x + 3} } {-1} + C\)

putting $a \gets -1$, $b \gets 3$

\(\ds \)

\(=\)

\(\ds \map \cos {3 - x} + C\)

simplifying

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XIV}$: $1$.