Primitive of Sine of a x by Cosine of b x

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Theorem

$\ds \int \sin a x \cos b x \rd x = \frac {-\map \cos {a - b} x} {2 \paren {a - b} } - \frac {\map \cos {a + b} x} {2 \paren {a + b} } + C$

for $a, b \in \R: a \ne b$


Proof

\(\ds \int \sin a x \cos b x \rd x\) \(=\) \(\ds \int \paren {\dfrac {\map \sin {a x + b x} + \map \sin {a x - b x} } 2} \rd x\) Werner Formula for Sine by Cosine
\(\ds \) \(=\) \(\ds \frac 1 2 \int \map \sin {a - b} x \rd x + \frac 1 2 \int \map \sin {a + b} x \rd x\) Linear Combination of Primitives
\(\ds \) \(=\) \(\ds \frac 1 2 \frac {-\map \cos {a - b} x} {a - b} - \frac 1 2 \frac {\map \cos {a + b} x} {a + b} + C\) Primitive of $\cos a x$
\(\ds \) \(=\) \(\ds \frac {-\map \cos {a - b} x} {2 \paren {a - b} } - \frac {\map \cos {a + b} x} {2 \paren {a + b} } + C\) simplifying

$\blacksquare$


Also see


Sources

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