Definite Integration
Functional Equation + Definite
Grade 12
Question:
<p>If \(f(x)+f(1-x)=1\) for all \(x\), find \(\displaystyle\int_0^1 f(x)\,dx\). [JEE Main 2016]</p>
Step-by-Step Solution
Key Concept: Let I = \int_0^1 f(x)dx. King: I = \int_0^1 f(1-x)dx. Add: 2I = \int_0^1 [f(x)+f(1-x)]dx = \int_0^1 1 dx = 1 \to I = 1/2.
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<p>$I=\int_0^1 f(x)dx\stackrel{\text{King}}{=}\int_0^1 f(1-x)dx$</p>
<p>$2I=\int_0^1[f(x)+f(1-x)]dx=\int_0^1 1\,dx=1\Rightarrow I=\boxed{\frac{1}{2}}$</p>
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Correct Answer: A