∫ e^(ax) sin bx dx = ?


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Question Number 142176 by malwaan last updated on 27/May/21

$\int e^{a x} s i n b x d x = ?$
	Answered by Dwaipayan Shikari last updated on 27/May/21

	$s i n (b x) = \frac{e^{i b x} - e^{- i b x}}{2 i}$ $\int e^{a x} (\frac{e^{i b x} - e^{- i b x}}{2 i}) d x$ $= \frac{e^{a x}}{2 i} (\frac{1}{a + i b} e^{i b x} - \frac{e^{- i b x}}{a - i b})$ $= \frac{e^{a x}}{2 i} (\frac{a - i b}{a^{2} + b^{2}} e^{i b x} - \frac{a + i b}{a^{2} + b^{2}} e^{- i b x})$ $= e^{a x} (\frac{a s i n (b x)}{a^{2} + b^{2}} - \frac{b c o s (b x)}{a^{2} + b^{2}}) + C$
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