For an integer n, the integral $\int\limits_0^πe^{cos^2x}cos^3 (2n +1)\, x\, dx$ has the value

0

$\int\limits_0^πe^{cos^2x}cos^3 (2n +1)\, x\, dx=\int\limits_0^πe^{cos^2(π-x)}cos^3(2n+1)(π-x)dx$

$=\int\limits_0^πe^{cos^2x}cos^3(2n\,π+π-(2n+1)x)dx$

$=-\int\limits_0^πe^{cos^2x}cos^3(2n+1)x\,dx$

Hence 2 I = 0 ⇒ I = 0.

Hence (C) is the correct answer.