Find the value of the \(\rm \int_{-\pi }^{\pi }cosx \ dx\)

Question

Question

Answer 1

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 \(\rm \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\)

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.