Q84 / 150 · MHT CET 2023 (13 May Shift 1) · Mathematics

If $\mathrm{f}(x)=\left\{\begin{array}{ll}\mathrm{e}^{\cos x} \sin x & , \text { for }|x| \leq 2 \\ 2, & \text { otherwise }\end{array}\right.$, then $\int_{-2}^3 \mathrm{f}(x) \mathrm{d} x$ is equal to

  • A
    0
  • Correct answer
    B
    2
  • C
    1
  • D
    3

Solution

$$ \begin{aligned} \int_{-2}^3 \mathrm{f}(x) \mathrm{d} x & =\int_{-2}^2 \mathrm{f}(x) \mathrm{d} x+\int_2^3 \mathrm{f}(x) \mathrm{d} x \\ & =\int_{-2}^2 \mathrm{e}^{\cos x} \sin x \mathrm{~d} x+\int_2^3 2 \mathrm{~d} x \end{aligned} $$ Since $\mathrm{e}^{\cos x} \sin x$ is an odd function. $$ \therefore \quad \int_{-2}^3 \mathrm{f}(x) \mathrm{d} x=0+2(3-2)=2 $$

Hence, the correct answer is option: 'B'.

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