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Integration as Antiderivative

Find the valu...

Question

Find the value of $\int_{- 1}^{1} [x^{2} + {x}] d x$ , where $[\cdot]$ and ${\cdot}$ denote the greatest function and fractional parts of $x$ , respectively.

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Solution

A] To find period of
$f (x) = s i n 3 x c o s [3 x] - c o s 3 x s i n [3 x]$
As
$s i n (A - B) = s i n A c o s B - c o s A s i n B$
So, $s i n 3 x c o s [3 x] - c o s 3 x s i n [3 x]$
$= s i n (3 x - [3 x])$
As $x - [x] = {x}$
So, $s i n (3 x - [3 x])$
$= s i n ({3 x})$
The period of fractional part of
x is 0 to 1
so, period of
$f (x) = s i n {3 x}$
is $\frac{1}{3}$

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