Indefinite Integration
Integration by Parts
Grade 12
Question:
<p>If an anti-derivative of <span class="math">\(f(x)\)</span> is <span class="math">\(e^x\)</span> and that of <span class="math">\(g(x)\)</span> is <span class="math">\(\cos x\)</span>, then <span class="math">\(\int f(x)\cos x \, dx + \int g(x) e^x \, dx\)</span> is equal to</p>
<p>(a) <span class="math">\(f(x) \times g(x) + C\)</span></p>
<p>(b) <span class="math">\(f(x) + g(x) + C\)</span></p>
<p>(c) <span class="math">\(e^x \cos x + C\)</span></p>
<p>(d) <span class="math">\(f(x) - g(x) + C\)</span></p>
Step-by-Step Solution
Key Concept: Recognize that if F(x) is an anti-derivative of f(x), then f(x) = F'(x). Use integration by parts strategically.
<p><strong>Solution:</strong> If anti-derivative of <span class="math">$f(x)$</span> is <span class="math">$e^x$</span>, then <span class="math">$f(x) = e^x$</span>. If anti-derivative of <span class="math">$g(x)$</span> is <span class="math">$\cos x$</span>, then <span class="math">$g(x) = -\sin x$</span>. Thus <span class="math">$\int e^x \cos x \, dx + \int (-\sin x) e^x \, dx = \int e^x(\cos x - \sin x) \, dx = e^x \cos x + C$</span>.</p>
Correct Answer: C