2.13: Exponential integral and curve from a derivative

0606/21/M/J/19 — Question 12 · 13 marks

(a)
Given that ∫0ae2x dx=50\displaystyle\int_{0}^{a} \mathrm{e}^{2x}\,\mathrm{d}x = 50, find the exact value of aa. You must show all your working.
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(b)(i)
A curve is such that dydx=3−2cos⁡5x\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3 - 2\cos 5x. The curve passes through the point (π5,8π5)\left(\tfrac{\pi}{5}, \tfrac{8\pi}{5}\right). Find the equation of the curve.
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(b)(ii)
Find ∫y dx\displaystyle\int y\,\mathrm{d}x and hence evaluate ∫π/2πy dx\displaystyle\int_{\pi/2}^{\pi} y\,\mathrm{d}x.
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