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Question Number 169253 by Mastermind last updated on 27/Apr/22

\int \frac{- s i n x}{e^{x}} d x

M a s t e r m i n d

Answered by thfchristopher last updated on 27/Apr/22

= - \int e^{- x} \sin x d x

= \int \sin x d (e^{- x})

= e^{- x} \sin x - \int e^{- x} d (\sin x)

= e^{- x} \sin x - \int e^{- x} \cos x d x

= e^{- x} \sin x + \int \cos x d (e^{- x})

= e^{- x} \sin x + e^{- x} \cos x - \int e^{- x} d (\cos x)

= e^{- x} (\sin x + \cos x) + \int e^{- x} \sin x d x

∴ - 2 \int e^{- x} \sin x d x = e^{- x} (\sin x + \cos x)

- \int e^{- x} \sin x d x = \frac{1}{2} e^{- x} (\sin x + \cos x) + C

Commented by Mastermind last updated on 28/Apr/22

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