If ∫_0^a[f(x)+f(-x)] d x=∫_ -a ^a g(x) d x then what is g(x) equal to?

Correct answer: A. f(x) ∫_0^a[f(x)+f(-x)] · d x=∫_ -a ^a g(x) · d x aligned & =∫_ -a ^0 g(x) · d x+∫_0^a g(x) · d x & =∫_0^a g(-x) · d x+∫_0^a g(x) · d x aligned =∫_0^a[g(x)+g(-x)] · d x f(x)=g(x)

NDA 2021

Exam: NDA
Year: 2021
Subject: Mathematics
Chapter: Definite Integration

NDA 2021 · Mathematics · Definite Integration

If ∫_0^a[f(x)+f(-x)] d x=∫_ -a ^a g(x) d x then what is g(x) equal...

\int_0^a[f(x)+f(-x)] d x=\int_{-a}^a g(x) d x

then what is

g(x)

equal to?

Options

A $f(x)$
B $f(-x)+f(x)$
C $-f(x)$
DNone of the above

Correct answer

A. $f(x)$

Step-by-step solution

\int_0^a[f(x)+f(-x)] \cdot d x=\int_{-a}^a g(x) \cdot d x

\begin{aligned} & =\int_{-a}^0 g(x) \cdot d x+\int_0^a g(x) \cdot d x \\ & =\int_0^a g(-x) \cdot d x+\int_0^a g(x) \cdot d x\end{aligned}

=\int_0^a[g(x)+g(-x)] \cdot d x\therefore f(x)=g(x)

Practice Definite Integration on Quantrex Academy →

More from Definite Integration

Passage: Let f(x)= x and g(x)-f(x)=f(4-x) . Question: What is ∫_0^4(f(x))/(g(x))\,dx… NDA 2026 Passage: Let ∫_0^ (π)/(2) (a x+b x)/((a+b)( x+ x))\,dx=k . Question: What is the value of… NDA 2026 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… NDA 2026 Passage: Let ∫_0^ (π)/(2) (a x+b x)/((a+b)( x+ x))\,dx=k . Question: What is ∫_0^ (π)/(2)… NDA 2026 Consider the following for the two (02) items that follow: Let f(x) = [x^2] , where [·]… NDA 2025 What is ∫_n^ n+1 (x - [x])\, dx , where [·] is the greatest integer function and n is a… NDA 2025 If I_1 = ∫_e^ e^2 (dx)/( x) and I_2 = ∫_1^2 (e^x)/(x)\, dx , then which one of the… NDA 2025 Paragraph: Let I=∫_0^ (π)/(2) (f(x))/(g(x)) d x , where f(x)= x and g(x)= x+ x+1 .… NDA 2024 Paragraph: Let f(t)= (t+ 1+t^2 ) and g(t)= (f(t)) . Question: What is ∫_ -π ^π g(t) d t… NDA 2024 Paragraph: Let f(x)=|x^2-x-2| Question: What is ∫_0^2 f(x) d x equal to? NDA 2024 What is ∫_ -1 ^1(3 x- 3 x) ^2 x d x equal to? NDA 2024 What is ∫_0^ (π)/(2) (a+ x)/(2 a+ x+ x) d x equal to? NDA 2024 Let p=∫_a^b f(x) d x and q=∫_a^b|f(x)| d x . If f(x)=e^ -x , then which one of the… NDA 2024 Let f(x)=|x-1|, g(x)=[x] and h(x)=f(x) g(x) where [.] is greatest integer function. What… NDA 2024 Let f(x)=|x-1|, g(x)=[x] and h(x)=f(x) g(x) where [.] is greatest integer function. What… NDA 2024 Let 3 f(x)+f((1)/(x))=(1)/(x)+1 What is 8 ∫_1^2 f(x) d x equal to? NDA 2024 Full Definite Integration list Definite Integration JEE Maths PYQs JEE Mathematics PYQs All NDA PYQs