The value of I = ∫ - 1 1 x sin ⁡ ( π x ) d x is (where [·] denotes the greatest integer function)

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Question: Answered & Verified by Expert

The value of

I = \int_{- 1}^{1} [x \sin (π x)] d x

is (where [·] denotes the greatest integer function)

MathematicsDefinite IntegrationJEE Main

Options:

A $π$
B $2 π$
C $0$
D $- π$

Solution:

2129 Upvotes Verified Answer

The correct answer is:

0

I = 2 \int_{0}^{1} [x \sin (π x)] d x

Now,

x \sin (π x) \in (0, 1)

as

x \in (0, 1)

∴ [x \sin (π x)] = 0

∴ I = 2 \int_{0}^{1} 0 d x = 0

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