Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\mathrm{e}^{-2x}(\cos x+2\sin x)\right]=-5\mat

Question

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\mathrm{e}^{-2x}(\cos x+2\sin x)\right]=-5\mathrm{e}^{-2x}\sin x$.[3] Hence evaluate the exact value of $\displaystyle\int_0^{\pi/4}\mathrm{e}^{-2x}\sin x\,\mathrm{d}x$.[4]

Accepted Answer

$\dfrac15\left(1-\dfrac{3\sqrt2}{2}\mathrm e^{-\pi/2}\right)$

Exponential and trigonometric product

Exponential and trigonometric product

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