Question

if $\ f ( x )$ $=\bigg \{ \begin {array} \ e^{\cos x}\sin \ x \\ 2 \\ \end {array} \begin {array} \ for |x|\le 2 \\ \text{otherwise} \\ \end {array}$ then $\int^{3}_{-2} f ( x ) \ dx$ is equal to

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is C

if $\ f ( x )$ $=\bigg \{ \begin {array}
\ e^{\cos x}\sin \ x \\
2 \\
\end {array} \begin {array}
\ for |x|\le 2 \\
\text{otherwise} \\
\end {array}$
$ \therefore \int^{3}_{-2} \ f ( x) \ dx = \int^2_{-2} f ( x ) \ dx + \int^3_2 \ f ( x ) \ dx $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \int^2_{-2} \ e^{\cos \ x } \ \sin \ x \ dx + \int^3_2 2 \ dx $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 0 + 2 [x]^3_2 $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [ \because e^{\cos \ x }\sin \ x $ is an odd function $] $
$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = 2 [ 3 - 2 ] = 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [ \because \int^{3}_{-2} \ f ( x ) \ dx = 2 ] $