Definite And Indefinite Integrals

We are given the condition:

%5Cint_0%5E%7B2a%7D%20f(x)%20%5C%2C%20dx%20%3D%202%20%5Cint_0%5Ea%20f(x)%20%5C%2C%20dx

Step 1: Use the property of definite integrals
The integral from 0 to

2a

can be split into two integrals:

%5Cint_0%5E%7B2a%7D%20f(x)%20%5C%2C%20dx%20%3D%20%5Cint_0%5Ea%20f(x)%20%5C%2C%20dx%20%2B%20%5Cint_a%5E%7B2a%7D%20f(x)%20%5C%2C%20dx

Step 2: Set up the equation
Substitute the given condition into this equation:

2%20%5Cint_0%5Ea%20f(x)%20%5C%2C%20dx%20%3D%20%5Cint_0%5Ea%20f(x)%20%5C%2C%20dx%20%2B%20%5Cint_a%5E%7B2a%7D%20f(x)%20%5C%2C%20dx

Simplifying this gives:

%5Cint_a%5E%7B2a%7D%20f(x)%20%5C%2C%20dx%20%3D%20%5Cint_0%5Ea%20f(x)%20%5C%2C%20dx

Step 3: Conclusion
Thus,

f(2a%20-%20x)%20%3D%20-f(x)

, which means

f(x)

is an odd function.
Hence, the correct answer is option (a).

Chapter Questions
1 MCQs