Passage: Let f(x)= x and g(x)-f(x)=f(4-x) . Question: What is ∫_0^4(f(4-x))/(g(4-x))\,dx equal to ?

Correct answer: C. 2 Given g(x) - f(x) = f(4-x) g(x) = f(x) + f(4-x) Replacing x with 4-x , we get: g(4-x) = f(4-x) + f(x) = g(x) Let I = ∫_0^4(f(4-x))/(g(4-x))\,dx Using the property ∫_0^a h(x)\,dx = ∫_0^a h(a-x)\,dx , we have: I = ∫_0^4(f(x))/(g(x))\,dx Adding the two expressions for I : 2I = ∫_0^4(f(4-x))/(g(4-x))\,dx + ∫_0^4(f(x))/(g(x))\,dx Since g(4-x) = g(x) , this becomes: 2I = ∫_0^4(f(4-x) + f(x))/(g(x))\,dx Substituting f(x) + f(4-x) = g(x) : 2I = ∫_0^4(g(x))/(g(x))\,dx = ∫_0^4 1\,dx 2I = [x]_0^4 = 4 I = 2 Answer: 2

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 x

and

g(x)-f(x)=f(4-x)

.

Question: What is

\displaystyle\int_0^4\dfrac{f(4-x)}{g(4-x)}\,dx

equal to ?

Options

A $0$
B $1$
C $2$
D $4$

Correct answer

C. $2$

Step-by-step solution

Given

g(x) - f(x) = f(4-x)

g(x) = f(x) + f(4-x)

Replacing

x

with

4-x

, we get:

g(4-x) = f(4-x) + f(x) = g(x)

Let

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

Using the property

\displaystyle\int_0^a h(x)\,dx = \displaystyle\int_0^a h(a-x)\,dx

, we have:

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

Adding the two expressions for

I

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

Since

g(4-x) = g(x)

, this becomes:

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

Substituting

f(x) + f(4-x) = g(x)

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

2I = [x]_0^4 = 4

I = 2

Answer:

2

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