If n is any integer, then _0^ e ^ ^2 x ^3(2 n +1) x d x= - Vidyadip

MCQ

If n is any integer, then $\int_0^\pi e ^{\cos ^2 x} \cos ^3(2 n +1) x d x=$

A
x
B
1
✓
$0$
D
None of these

✓

Answer

Correct option: C.

$0$

(C)
Let $f (x)= e ^{\cos ^2 x} \cos ^3(2 n +1) x$
$\therefore \quad f (\pi-x)= e ^{\cos ^2(\pi-x)} \cos ^3[(2 n +1)(\pi-x)]$
$= e ^{\cos ^2 x} \cos ^3[(2 n +1) \pi-(2 n +1) x]$
$=- e ^{\cos ^2 x} \cos ^3(2 n +1) x=- f (x)$
Since $\int_0^{2 a } f (x) d x=0$, if $f (2 a -x)=- f (x)$
$\therefore \quad \int_0^\pi e ^{\cos ^2 x} \cos ^3(2 n +1) x d x=0$

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