Question
If $$f\left( x \right)$$ is an even function, then what is $$\int\limits_0^\pi {f\left( {\cos \,x} \right)} dx$$ equal to?
A.
$$0$$
B.
$$\int\limits_0^{\frac{\pi }{2}} {f\left( {\cos \,x} \right)} dx$$
C.
$$2\int\limits_0^{\frac{\pi }{2}} {f\left( {\cos \,x} \right)} dx$$
D.
$$1$$
Answer :
$$2\int\limits_0^{\frac{\pi }{2}} {f\left( {\cos \,x} \right)} dx$$
Solution :
Since $$f\left( x \right)$$ is an even function therefore $$\int\limits_0^\pi {f\left( x \right)} dx = 2\int\limits_0^{\frac{\pi }{2}} {f\left( x \right)} dx$$
Hence, $$\int\limits_0^\pi {f\left( {\cos \,x} \right)} dx = 2\int\limits_0^{\frac{\pi }{2}} {f\left( {\cos \,x} \right)} dx$$