JEE Main 2023 · Mathematics · Indefinite and Definite Integrals · medium

The value of 16∫0^1 x [4x] dx, where [ · ] denotes the greatest integer function, is

Numerical answer

The value of

16\displaystyle\int_0^1 x\,[4x]\,dx

, where

[\,\cdot\,]

denotes the greatest integer function, is

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Answer: 17

\left[\frac k4,\frac{k+1}4\right)

we have

[4x]=k

for

k=0,1,2,3

, so the integral splits into four pieces

k\int_{k/4}^{(k+1)/4}x\,dx=k\cdot\frac{(k+1)^2-k^2}{32}=\frac{k(2k+1)}{32}

. Evaluating:

k=0

gives

0

k=1

gives

\frac3{32}

k=2

gives

\frac{10}{32}

k=3

gives

\frac{21}{32}

. The sum is

\frac{34}{32}

, so the integral equals

\frac{17}{16}

and

16

times it is

17

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