Primitive of Exponential of a x by Sine of b x/Lemma
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Lemma for Primitive of $e^{a x} \sin b x$
- $\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x$
Proof
With a view to expressing the primitive in the form:
- $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$
let:
\(\ds u\) | \(=\) | \(\ds \sin b x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d u} {\d x}\) | \(=\) | \(\ds b \cos b x\) | Derivative of $\sin a x$ |
and let:
\(\ds \frac {\d v} {\ d x}\) | \(=\) | \(\ds e^{a x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds v\) | \(=\) | \(\ds \frac {e^{a x} } a\) | Primitive of $e^{a x}$ |
Then:
\(\ds \int e^{a x} \sin b x \rd x\) | \(=\) | \(\ds \sin b x \paren {\frac {e^{a x} } a} - \int \paren {\frac {e^{a x} } a} \paren {b \cos b x} \rd x + C\) | Integration by Parts | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x\) | Primitive of Constant Multiple of Function |
$\blacksquare$