e-ax-sin-bx-dx-

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