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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Definite Integration

NDA 2021 · Mathematics · Definite Integration

What is ∫_0^a (f(a-x))/(f(x)+f(a-x)) d x equal to?

What is ∫0af(a−x)f(x)+f(a−x)dx\int_0^a \frac{f(a-x)}{f(x)+f(a-x)} d x equal to?

Options

  1. Aaa
  2. B2a2 a
  3. C00
  4. Da2\frac{a}{2}

Correct answer

D. a2\frac{a}{2}

Step-by-step solution

I=∫0af(a−x)f(x)+f(a−x)⋅dx=∫0af(a+0−a+x)f(a+0−x)+f(a+0−a+x)⋅dx\begin{aligned} I & =\int_0^a \frac{f(a-x)}{f(x)+f(a-x)} \cdot d x \\ & =\int_0^a \frac{f(a+0-a+x)}{f(a+0-x)+f(a+0-a+x)} \cdot d x\end{aligned}=∫0af(x)f(a−x)+f(x)⋅dx2I=∫0af(a−x)+f(x)f(a−x)+f(x)⋅dx=[x]0a\begin{aligned} & =\int_0^a \frac{f(x)}{f(a-x)+f(x)} \cdot d x \\ & 2 I=\int_0^a \frac{f(a-x)+f(x)}{f(a-x)+f(x)} \cdot d x=[x]_0^a\end{aligned}\therefore I=\frac{a}{2}$
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