Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>The value of <math>∫<sup>π/2</sup><sub>0</sub> \frac{\sin x}{\sin x + \cos x} dx</math> is</p>
<p>(a) <math>\frac{π - 1}{2}</math></p>
<p>(b) <math>\frac{π}{8}</math></p>
<p>(c) <math>\frac{π - 2}{4}</math></p>
<p>(d) <math>\frac{π - 1}{4}</math></p>
Step-by-Step Solution
Key Concept: Use the symmetry property of integrals: if I = ∫ f(x) dx, then also I = ∫ f(a-x) dx over the same interval, allowing you to add them.
<p>To evaluate this integral, use the property that <math>∫<sup>a</sup><sub>0</sub> f(x) dx = ∫<sup>a</sup><sub>0</sub> f(a-x) dx</math>. Let I = <math>∫<sup>π/2</sup><sub>0</sub> \frac{\sin x}{\sin x + \cos x} dx</math>. Then also I = <math>∫<sup>π/2</sup><sub>0</sub> \frac{\cos x}{\sin x + \cos x} dx</math>. Adding these: 2I = <math>∫<sup>π/2</sup><sub>0</sub> 1 dx = π/2</math>. Therefore I = <math>\frac{π}{4}</math>.</p>
Correct Answer: C