Calculate the following Integral $\int_{0}^{\infty} \frac{|\sin(\pi*x)|}{\lfloor x \rfloor} dx$

Question

So I have to calculate the following Integral: $\int_{0}^{\infty} \frac{|\sin(\pi*x)|} {\lfloor x \rfloor} dx$

I know how to calculate improper Integrals, but I really do have a problem with the $\lfloor x \rfloor$ (everytime it comes to any analysis excercise i can't solve it) even though i have read it's definition a lot of times and as far as i understand it it would be the greatest integer less than or equal to x. I also have a problem with he absolute value when it comes to integrals.

I would like to add up here my part of the solution but in this case i really am stuck and have no clue what to do. Therefore i would really be greatful for any sort of help.

Thanks in advance

Answer 1

I will solve the integral $$\int_1^{+ \infty} \frac{|\sin ( \pi x)|}{\lfloor x \rfloor} \mathrm d x$$ since for $0 < x < 1$ you have something divided by zero.

Split the integral into a series: $$\int_1^{+ \infty} \frac{|\sin ( \pi x)|}{\lfloor x \rfloor} \mathrm d x = \sum_{n=1}^{+ \infty} \int_n^{n+1} \frac{|\sin ( \pi x)|}{\lfloor x \rfloor} \mathrm d x $$ In the interval $(n,n+1)$ the floor function is constant and evaluates $n$.

Moreover $|\sin ( \pi x)|$ is a periodic function with period $1$. In particular $$\int_n^{n+1} |\sin ( \pi x)| \mathrm d x = \int_0^1 |\sin ( \pi x)| \mathrm d x = C$$ where $C$ is a positive constant I don't want to compute.

Hence $$\sum_{n=1}^{+ \infty} \int_n^{n+1} \frac{|\sin ( \pi x)|}{\lfloor x \rfloor} \mathrm d x= \sum_{n=1}^{+ \infty} \int_n^{n+1} \frac{|\sin ( \pi x)|}{n} \mathrm d x= \sum_{n=1}^{+ \infty} \frac{C}{n} = + \infty $$ So the integral is divergent.