Primitive of Sine of a x + b/Proof 2
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Corollary to Primitive of Sine Function
- $\ds \int \map \sin {a x + b} \rd x = -\frac {\map \cos {a x + b} } a + C$
Proof
Let $u = a x + b$.
Then:
- $\dfrac {\d u} {\d x} = a$
Then:
\(\ds \int \map \sin {a x + b} \rd x\) | \(=\) | \(\ds \int \dfrac {\sin u} a \rd u\) | Integration by Substitution | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac {\cos u} a\) | Primitive of $\sin u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {\map \cos {a x + b} } a + C\) | substituting back for $u$ |
$\blacksquare$
Sources
- 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 8$. Change of Variable