<p>The integral <span class="math">\int (x\sin x + \cos x)dx</span> is equal to</p>
<p>(a) <span class="math">\frac{x\cos x}{x\sin x + \cos x} + \tan x + C</span></p>
<p>(b) <span class="math">-\frac{x\cos x}{x\sin x + \cos x} + \tan x + C</span></p>
<p>(c) <span class="math">-x\sin x + \cos x + \cot x + C</span></p>
<p>(d) None of the above</p>
Step-by-Step Solution
Key Concept: Use integration by parts for products of polynomial and trigonometric functions, combining results from multiple terms.
<p><strong>Step 1:</strong> Integrate each term: <span class="math">\int (x\sin x + \cos x)dx = \int x\sin x\, dx + \int \cos x\, dx</span></p><p><strong>Step 2:</strong> For <span class="math">\int x\sin x\, dx</span>, use integration by parts with <span class="math">u = x</span>, <span class="math">dv = \sin x\, dx</span>: <span class="math">= -x\cos x + \int \cos x\, dx</span></p><p><strong>Step 3:</strong> <span class="math">= -x\cos x + \sin x + \sin x + C = -x\cos x + 2\sin x + C</span></p><p>∴ Answer is (b).</p>
Correct Answer: b