$\int_{0}^{5} \cos \left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right) d x$, Where [ $t$ ] denotes greatest integer less than or equal to $t$, is equal to :

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Definite Integrals

\int_{0}^{5} cos (π (x - [\frac{x}{2}])) d x

,
Where [

t

] denotes greatest integer less than or equal to

t

, is equal to :

Answer: (D)

We split the integral based on the greatest integer function:

I = \int_{0}^{2} cos (π x) d x + \int_{2}^{4} cos (π x - π) d x + \int_{4}^{5} cos (π x - 2 π) d x

Note that

cos (π x - π) = - cos (π x)

and

cos (π x - 2 π) = cos (π x)

I = [\frac{sin π x}{π}]_{0}^{2} + [\frac{sin ( π x - π )}{π}]_{2}^{4} + [\frac{sin ( π x - 2 π )}{π}]_{4}^{5}

Since

sin (nπ) = 0

for all integers

n

, each term evaluates to 0.

Therefore,

I = 0

JEE Main · 2022 · 29 Jun 2022 · Shift 1 · Single correct option

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