The value of $\int_{- 1}^{1}{\lbrack x\lbrack 1 + \sin{\pi x\rbrack\ dx}}$ is ($\lbrack \bullet \rbrack$ denotes the greatest integer)

Question

Dr. Abubacker · Accepted Answer

(a)
$\int_{- 1}^{1}{\lbrack x\lbrack 1 + \sin{\pi xbrack + 1brack dx}}$
$= \int_{- 1}^{0}{\lbrack x\lbrack 1 + \sin{\pi xbrack + 1brack dx + \int_{0}^{1}{\lbrack x\lbrack 1 + \sin{\pi xbrack + 1brack dx}}}}$
Now, $- 1 < x < 0\ \  \Rightarrow \ \lbrack 1 + \sin{\pi xbrack = 0}$
$0 < x < 1\ \  \Rightarrow \ \lbrack 1 + \sin{\pi xbrack = 1}$
$\Rightarrow \ \lbrack x\lbrack 1 + \sin{\pi xbrack + 1brack = 1}$
$	herefore\ \int_{- 1}^{1}{\lbrack x\lbrack 1 + \sin{\pi xbrack + 1brack dx = 2}}$

Q. The value of $\int_{- 1}^{1}{\lbrack x\lbrack 1 + \sin{\pi x\rbrack\ dx}}$ is ($\lbrack \bullet \rbrack$ denotes the greatest integer)

Explanation

Reviewed by Dr. Abubacker

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