For any integer n the integral ∫π0ecos2xcos3(2n+1)xdx has the value
A
π
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B
1
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C
0
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D
None of these
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Solution
The correct option is C0 I=∫π0ecos2xcos3(2n+1)xdx,nϵZ...(1)=∫π0ecos2(π−x)cos3[(2n+1)(π−x)]dxUsing∫a0f(x)dx=∫a0f(a−x)dx∴I=∫π0ecos2xcos3[(2n+1)π−(2n+1)x]dxI=∫π0−ecos2xcos3(2n+1)xdx...(2) Adding (1) and (2) we get 2I=0⇒I=0