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Question: If \(f ( x ) = \left\{ \begin{array} { c c } e ^ { \cos x } \sin x , & | x | \leq 2 \\ 2 , & \text {...

If f(x)={ecos⁡xsin⁡x,∣x∣≤22, otherwise f ( x ) = \left\{ \begin{array} { c c } e ^ { \cos x } \sin x , & | x | \leq 2 \\ 2 , & \text { otherwise } \end{array} \right., then ∫−23f(x)dx\int _ { - 2 } ^ { 3 } f ( x ) d x is equal to

A

0

B

1

C

2

D

3

Answer

2

Explanation

Solution

∫−23f(x)dx=∫−22f(x)dx+∫23f(x)dx\int _ { - 2 } ^ { 3 } f ( x ) d x = \int _ { - 2 } ^ { 2 } f ( x ) d x + \int _ { 2 } ^ { 3 } f ( x ) d x

ecos⁡xsin⁡xe ^ { \cos x } \sin xis an odd function

∴∫−23f(x)dx=∫−22ecos⁡xsin⁡xdx+∫232dx=0+2(3−2)=2\therefore \int _ { - 2 } ^ { 3 } f ( x ) d x = \int _ { - 2 } ^ { 2 } e ^ { \cos x } \sin x d x + \int _ { 2 } ^ { 3 } 2 d x = 0 + 2 ( 3 - 2 ) = 2 .