# Calculation of value of $\int_{0}^{n}\cos\left(\lfloor x \rfloor\cdot \{x\}\right)dx\;,$

$$(1)$$ Calculation of value of $$\displaystyle \int_{0}^{n}\cos\left(\lfloor x \rfloor\cdot \{x\}\right)dx\;,$$ Where $$\lfloor x \rfloor$$ is floor function of $$x$$ and $$\{x\} = x-\lfloor x \rfloor.$$ and $$n$$ is a positive integer.

$$(2)\;$$Calculation of least positive integer $$n$$ for which $$\displaystyle \int_{1}^{n}\lfloor x \rfloor \cdot \lfloor \sqrt{x} \rfloor dx>60\;,$$ Where $$\lfloor x \rfloor$$ is floor

function of $$x$$

$$\bf{My\; Try::}$$ For $$(1)$$ one:: We can write $$\displaystyle \int_{0}^{n}\cos\left(\lfloor x \rfloor\cdot \{x\}\right)dx = \displaystyle \int_{0}^{n}\cos\left(\lfloor x \rfloor\cdot \left(x-\lfloor x \rfloor\right)\right)dx$$

So $$\displaystyle \int_{0}^{n}\cos\left(\lfloor x \rfloor\cdot \left(x-\lfloor x \rfloor\right)\right)dx = \int_{0}^{1}\cos(0)dx+\int_{1}^{2}\cos(1\cdot (x-1))dx+\int_{2}^{3}\cos(2\cdot(x-2))dx+..........+\int_{n-1}^{n}\cos((n-1)(x-(n-1)))dx$$

Now How can I solve after that, Help me, Thanks

$$\bf{My\; Try}::$$ For $$(2)$$ one:: We can write

$$\displaystyle \int_{1}^{n}\lfloor x \rfloor \cdot \lfloor \sqrt{x} \rfloor dx = \int_{0}^{1}0\cdot 0dx+\int_{1}^{2}1\cdot 1dx+\int_{2}^{3}2\cdot 1dx++\int_{3}^{4}3\cdot 1dx+......+\int_{n-1}^{n}(n-1)\lfloor \sqrt{n-1}\rfloor x$$

Now How can i solve after that, Help me, Thanks

• Have you tried substituting $y=k(x-k)$ and evaluating the integrals? Nov 2, 2014 at 16:53

Hint: For $k \geq 1$,
$$\int_k^{k+1} \cos(k(x-k)) \ dx = \int_0^1 \cos(kx) \ dx = \frac{1}{k} \sin(k)$$