Mathematics

If $$\displaystyle f\left ( x \right )= \left\{\begin{matrix}e^{\cos x}\cdot \sin x &for\left | x \right |\leq 2 \\2  &otherwise \end{matrix}\right.$$ then $$\displaystyle \int_{-2}^{3}f\left ( x \right )dx$$ is equal to 


ANSWER

2


SOLUTION
Given $$f\left( x \right) =\begin{cases} { e }^{ \cos { x }  }\sin { x } ,\quad \left| x \right| \le 2 \\ 2\quad \quad \quad \quad \quad \quad else \end{cases}$$
$$\therefore \int _{ -2 }^{ 2 }{ f\left( x \right)  } dx+\int _{ 2 }^{ 3 }{ f\left( x \right)  } dx=\int _{ -2 }^{ 2 }{ { e }^{ \cos { x }  }\sin { x } dx } +\int _{ 2 }^{ 3 }{ 2 } dx$$
$$=0+2\left[ x \right] _{ 2 }^{ 3 }$$  ($$\because { e }^{ \cos { x }  }\sin { x } $$ is an odd function)
$$=2\left[ 3-2 \right] =2\quad \left[ \because \int _{ -2 }^{ 3 }{ f\left( x \right)  } dx=2 \right] $$
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Single Correct Medium Published on 17th 09, 2020
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