3.49: Product rule for x sin x, tangent and integral

0606/21/O/N/22 — Question 8 · 10 marks

The equation of a curve is y=xsin⁡xy = x\sin x.
(a)
Find dydx\dfrac{\mathrm{d}y}{\mathrm{d}x}.
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(b)
Find the equation of the tangent to the curve at x=π2x = \dfrac{\pi}{2} in the form y=mx+cy = mx + c.
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(c)
Use your answer to (a) to find ∫xcos⁡x dx\displaystyle\int x\cos x\,\mathrm{d}x.
[3]
(d)
Evaluate ∫0π/4xcos⁡x dx\displaystyle\int_{0}^{\pi/4} x\cos x\,\mathrm{d}x, giving your answer correct to 22 significant figures.
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