Primitive of Sine of a x + b

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 $\map \sin {3 - x}$

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