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JEE Main 2000 Mathematics question on Definite Integrals and Areas

If f(x)={ecos⁡xsin⁡x, for ∣x∣≤22, otherwisef(x) = \begin{cases} e^{\cos x} \sin x & , \text{ for } |x| \leq 2 \\ 2 & , \text{ otherwise} \end{cases}

then ∫−23f(x) dx\int_{-2}^{3} f(x)\, dx is equal to
  1. A.
    0
  2. B.
    1
  3. C.
    2
  4. D.
    3
Show answer and solution

Correct option: C

Solution

Split the integral: ∫₋₂³ f(x)dx = ∫₋₂² e^(cosx)·sinx dx + ∫₂³ 2 dx.

For the first part, note that e^(cosx)·sinx is an odd function (replacing x with -x gives -e^(cosx)·sinx), and the interval [-2, 2] is symmetric about 0, so ∫₋₂² e^(cosx)·sinx dx = 0.

The second part: ∫₂³ 2 dx = 2·(3-2) = 2.

Therefore the total is 0 + 2 = 2.

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