Correct Answer - Option 2 : f(-x) = -f(x)
Concept:
1. The rules for integrating even and odd functions:
If the function is even or odd and the interval is [-a, a], we can apply these rules:
- When f(x) is even ⇔ \(\mathop \smallint \nolimits_{ - a}^a f\left( x \right)dx = 2\mathop \smallint \nolimits_0^a f\left( x \right)dx{\rm{\;}}\)
- When f(x) is odd ⇔ \(\mathop \smallint \nolimits_{ - a}^a f\left( x \right)dx = 0\)
2. If f is even then f (-x) = f (x)
3. If f is odd then f (-x) = -f (x)
Calculation:
Given that
\(\int_{-1}^{1}f(x)dx= 0 \)
According to the fundamental property discussed above, this will be possible only if
f(-x) = -f(x)
