Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>Evaluate \(\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx\)</p>
<p>(a) 0</p>
<p>(b) \(\frac{1}{4}\)</p>
<p>(c) \(\frac{\pi}{2}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the property that $\int_0^a f(x)dx = \int_0^a f(a-x)dx$ and add complementary integrals to simplify.
<p><strong>Solution:</strong> Using the property $\int_0^a f(x)dx = \int_0^a f(a-x)dx$, we add the original integral with its complementary form.</p><p>Let $I = \int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$ ...(i)</p><p>Then $I = \int_0^{\pi/2} \frac{\sin(\frac{\pi}{2} - x)}{\sin(\frac{\pi}{2} - x) + \cos(\frac{\pi}{2} - x)} dx = \int_0^{\pi/2} \frac{\cos x}{\cos x + \sin x} dx$ ...(ii)</p><p>Adding (i) and (ii): $2I = \int_0^{\pi/2} \frac{\sin x + \cos x}{\sin x + \cos x} dx = \int_0^{\pi/2} 1 \, dx = \frac{\pi}{2}$</p><p>Therefore $I = \frac{\pi}{4}$</p>
Correct Answer: B