Primitive of Exponential of a x by Sine of b x/Lemma

Lemma for Primitive of $e^{a x} \sin b x$

$\ds \int e^{a x} \sin b x \rd x = \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x$

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 \ \ $

$\ds \frac {\d u} {\d x}$

$=$

$\ds b \cos b x$

and let:

$\ds \frac {\d v} {\ d x}$

$=$

$\ds e^{a x}$

$\ds \leadsto \ \ $

$\ds v$

$=$

$\ds \frac {e^{a x} } a$

Then:

$\ds \int e^{a x} \sin b x \rd x$

$=$

$\ds \sin b x \paren {\frac {e^{a x} } a} - \int \paren {\frac {e^{a x} } a} \paren {b \cos b x} \rd x + C$

$\ds $

$=$

$\ds \frac {e^{a x} \sin b x} a - \frac b a \int e^{a x} \cos b x \rd x$

$\blacksquare$