Correct Answer - Option 1 : 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: \(\rm \int_{-\pi }^{\pi }cosx \ dx\)
Let f(x) = cos x
As we can see that, f(- x) = cos (- x) = cos x = f(x).
So, cos x is an even function.
As we know that, when f(x) is an even function then \(\rm \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)
\(\Rightarrow \rm \int_{-\pi }^{\pi }cosx \ dx=2\int_{0 }^{\pi }cosx \ dx=2(sin\pi-sin0)=0\)
Hence, the correct option is 1.
