# Primitive of Exponential of a x by Sine of b x/Proof 2

$\displaystyle \int e^{a x} \sin b x \rd x = \frac {e^{a x} \paren {a \sin b x - b \cos b x} } {a^2 + b^2} + C$
 $\displaystyle \cos b x + i \sin bx$ $=$ $\displaystyle e^{i b x}$ Euler's Formula $\displaystyle \leadsto \ \$ $\displaystyle e^{a x} \cos b x + i e^{a x} \sin b x$ $=$ $\displaystyle e^{a x} e^{i b x}$ multiplying both sides by $e^{a x}$ $\displaystyle$ $=$ $\displaystyle e^{\left({a + i b}\right)x}$ Exponent Combination Laws $\displaystyle \leadsto \ \$ $\displaystyle \int e^{a x} \cos b x \, \mathrm d x + i \int e^{a x} \sin b x \rd x$ $=$ $\displaystyle \int e^{\left({a + i b}\right) x} \rd x$ Linear Combination of Complex Integrals $\displaystyle$ $=$ $\displaystyle \frac 1 {a + i b} e^{\paren {a + i b} x} + C$ Primitive of $e^{a x}$ $\displaystyle$ $=$ $\displaystyle \frac {a - i b} {a^2 + b^2} e^{\paren {a + i b} x} + C$ multiplying top and bottom by $a - i b$ $\displaystyle$ $=$ $\displaystyle \frac {a - i b} {a^2 + b^2} e^{a x} e^{i b x} + C$ Exponent Combination Laws $\displaystyle$ $=$ $\displaystyle \frac {a - i b} {a^2 + b^2} e^{a x} \paren {\cos b x + i \sin b x} + C$ Euler's Formula $\displaystyle$ $=$ $\displaystyle \frac a {a^2 + b^2} e^{a x} \cos b x - \frac {i b} {a^2 + b^2} e^{a x} \cos b x$ $\displaystyle$  $\, \displaystyle + \,$ $\displaystyle \frac {i a} {a^2 + b^2} e^{a x} \sin b x + \frac b {a^2 + b^2} e^{a x} \sin b x + C$
$\blacksquare$