Find the anti-derivative (or integral) of the following by the method of inspection.
$s i n 2 x - 4 e^{3 x} .$

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Solution

$= \int s i n 2 x d x - 4 \int e^{3 x} d x = \frac{- c o s 2 x}{2} - 4 \frac{e^{3 x}}{3} = - \frac{1}{2} c o s 2 x - \frac{4}{3} e^{3 x} (∵ \frac{d}{d x} {- \frac{1}{2} c o s 2 x} = - \frac{1}{2} (- s i n 2 x) 2) = s i n 2 x a n d \frac{d}{d x} (\frac{4}{3} (e^{3 x} 3)) = 4 e^{3 x}$

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