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NDA 2026
Exam
NDA
Year
2026
Subject
Mathematics
Chapter
Definite Integration

NDA 2026 · Mathematics · Definite Integration

Passage: Let f(x)= x and g(x)-f(x)=f(4-x) . Question: What is...

Passage: Let f(x)=sin⁡xf(x)=\sin x and g(x)−f(x)=f(4−x)g(x)-f(x)=f(4-x).

Question: What is ∫04f(x)g(x) dx\displaystyle\int_0^4\dfrac{f(x)}{g(x)}\,dx equal to ?

Options

  1. A00
  2. B11
  3. C22
  4. D44

Correct answer

C. 22

Step-by-step solution

I=∫04f(x)g(x) dxI = \displaystyle\int_0^4 \dfrac{f(x)}{g(x)} \,dx

From the given equation, g(x)=f(x)+f(4−x)g(x) = f(x) + f(4-x)

Using the property ∫0ah(x) dx=∫0ah(a−x) dx\displaystyle\int_0^a h(x) \,dx = \displaystyle\int_0^a h(a-x) \,dx:

I=∫04f(4−x)g(4−x) dxI = \displaystyle\int_0^4 \dfrac{f(4-x)}{g(4-x)} \,dx

Since g(4−x)=f(4−x)+f(x)=g(x)g(4-x) = f(4-x) + f(x) = g(x):

I=∫04f(4−x)g(x) dxI = \displaystyle\int_0^4 \dfrac{f(4-x)}{g(x)} \,dx

Adding the two equations for II:

2I=∫04f(x)+f(4−x)g(x) dx2I = \displaystyle\int_0^4 \dfrac{f(x) + f(4-x)}{g(x)} \,dx

2I=∫04g(x)g(x) dx=∫041 dx2I = \displaystyle\int_0^4 \dfrac{g(x)}{g(x)} \,dx = \displaystyle\int_0^4 1 \,dx

2I=4⇒I=22I = 4 \Rightarrow I = 2

Answer: 22
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