Primitive of Sine of a x + b
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Corollary to Primitive of Sine Function
- $\ds \int \map \sin {a x + b} \rd x = -\frac {\map \cos {a x + b} } a + C$
where $a$ is a non-zero constant.
Proof 1
\(\ds \int \sin x \rd x\) | \(=\) | \(\ds -\cos x + C\) | 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 |
$\blacksquare$
Proof 2
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 $u$ |
$\blacksquare$
Examples
Primitive of $\map \sin {3 x + 4}$
- $\ds \int \map \sin {3 x + 4} \rd x = -\dfrac {\map \cos {3 x + 4} } 3 + C$
Primitive of $\map \sin {3 - x}$
- $\ds \int \map \sin {3 - x} \rd x = \map \cos {3 - x} + C$