Evaluate $\int_{0}^{3\pi/4} [(1+x)\sin x + (1-x)\cos x] dx$
Step-by-Step Solution
Key Concept: General
Let $I = \int_{0}^{3\pi/4} [(1+x)\sin x + (1-x)\cos x] dx$<br>$= \int_{0}^{3\pi/4} ((\sin x + \cos x) + x(\sin x - \cos x)) dx$<br>Integrating by parts, we get<br>$= [(\sin x - \cos x) + x(-\cos x - \sin x) - \int (-\cos x - \sin x) dx]_0^{3\pi/4}$<br>$= [2(\sin x - \cos x) - x(\sin x + \cos x)]_0^{3\pi/4} = 2[\sqrt{2} + 1]$
Correct Answer: $2[\sqrt{2} + 1]$