Integrate Box Function?

Calculus Level 3

$\int_1^3 \lfloor x \rfloor \cos \left( \dfrac\pi2 \big( x - \lfloor x \rfloor \big) \right) \, dx$

The integral above is equal to $\frac a{b\pi}$ , where $a$ and $b$ are coprime positive integers.

Find $a+b$ .

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .

The answer is 7.

1 solution

Aditya Narayan Sharma
Feb 19, 2017

Relevant wiki: Integration of Piecewise Functions

$\displaystyle \int _{ 1 }^{ 3 }{ \left[ x \right] \cos { \left( \frac { \pi }{ 2 } \left( x-\left[ x \right] \right) \right) dx } } \\ =\int\limits_1^2 \cos\left(\dfrac{\pi}{2}(x-1)\right)\; dx+ 2\int\limits_2^3 \cos\left(\dfrac{\pi}{2}(x-2)\right)\\=\dfrac{2}{\pi}+\dfrac{4}{\pi} \\=\dfrac{6}{\pi}$

So , $a=6,b=1$ making the answer $6+1=\boxed{7}$

Ya. Very well done solution. Did the same way. But You must write that $a=6$ and $b=1$

Hence $a+b$ = $6+1 = \boxed{7}$ because the answer is $\boxed{7}$ but not $\dfrac{6}{\pi}$

Md Zuhair - 4 years, 3 months ago

I thought the reader would deduce this themselves, anyways added.

Aditya Narayan Sharma - 4 years, 3 months ago

Ya that anyone can deduce, But you know that $\dfrac{6}{\pi}$ is not the answer, where $\boxed{7}$ Is. So i just told. Anyways. Thanks

Md Zuhair - 4 years, 3 months ago

Integrate Box Function?

The answer is 7.

1 solution

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