Correct Answer - Option 2 : 0
Concept:
The function f(x) is an odd function if f(x) = - f(-x) and an even function if f(x) = f(-x).
- When f(x) is an even function then \(\rm \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)
- When f(x) is odd function then \(\rm \int_{-a}^{a}f(x)dx=0\)
Calculation:
Given: \(\rm \int_{-\pi }^{\pi }sinx \ dx\)
Let f(x) = sin x
As we know that, f(- x) = sin (- x) = - sin x = - f(x).
So, sin x is an odd function.
As we know that when f(x) is an odd function then \(\rm \int_{-a}^{a}f(x)dx=0\)
\(\Rightarrow\rm \int_{-\pi }^{\pi }sinx \ dx=0\)
Hence, the correct option is 2.
