integral and floor function*******

**Ajreb** · February 18th 2009, 07:15 PM

can you solve $\int_{0}^{1332}{\sin ^{2}\Big(\pi \big(x-\left\lfloor x \right\rfloor \big)\Big)\,dx}$

**Krizalid** · February 18th 2009, 07:21 PM

Originally Posted by Ajreb

can you solve

Yes.

Originally Posted by Ajreb

$\int_{0}^{1332}{\sin ^{2}\Big(\pi \big(x-\left\lfloor x \right\rfloor \big)\Big)\,dx}$

$\begin{aligned} \int_{0}^{1332}{\sin ^{2}\left( \pi \left( x-\left\lfloor x \right\rfloor \right) \right)\,dx}&=\sum\limits_{k=0}^{1331}{\int_{k}^{k +1}{\sin ^{2}\big(\pi (x-k)\big)\,dx}} \\ & =\frac{1}{\pi }\sum\limits_{k=0}^{1331}{\int_{0}^{\pi }{\sin ^{2}x\,dx}} \\ & =\frac{1}{\pi }\sum\limits_{k=0}^{1331}{\frac{\pi }{2}} \\ & =666. \end{aligned}$

Oh-oh, bad number.

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