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How can this be integrated?

$\displaystyle \int_0^{\pi/4} \left(\;\tan ^n x + \tan^{n-2} x \;\right) \;\; d\left(x-\cfrac{[x]}{1!}+\cfrac{[x]^2}{2!}-\cfrac{[x]^3}{3!} + \,...\right)$

where [.] denotes greatest integer function.

Welcome to MSE. Your question is phrased as an isolated problem.

Sorry for not providing any context along with the question, as I was working on my integrals, I came across this question and wondered about integrating with respect to $f(x)$ rather than $x$. Aforementioned series without $[x]$ looks quite similar to $ e^{-x}$.

I do realize that $ \displaystyle \int f(x) \; d\,g(x) = \displaystyle \int f(x) \;g'(x)\;dx $

Since the range is from $0$ to $\pi/4$, [x] would be $0$ but I am not sure if that's helpful, and that's all I was able to decipher.

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    $\begingroup$ I'm having a problem understanding your expression. What is the integrating variable? If the thing in the parentheses is the integrating variable (because d is in front of it), then $\tan^{n-2}$ has no argument. Otherwise, you are missing an integrating variable... you should clarify what this means. Also, any use of floor function will only split this integral into two parts - 0 to 1 and 1 to π/2. $\endgroup$
    – orion
    Jan 14, 2019 at 10:21
  • $\begingroup$ @orion Apologies for the missing variable, I have made the necessary corrections and added context to it. $\endgroup$ Jan 14, 2019 at 10:29

1 Answer 1

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The main thing to realize here is what the parenthesized expression means for different ranges in $x$:

$$u=x-\frac{\lfloor x\rfloor}{1!}+\frac{\lfloor x\rfloor^2}{2!}+\cdots$$

For $0\leq x<1$, the expression is simply equal to $x$.

For $1\leq x<2$, $\lfloor x \rfloor=1$. So you can write this as: $$u=x-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots=x-1+e^{-1}$$ where I recognized the value of $e^{-1}$ without the first term. For $x>2$, you need to reevaluate it again, you get $x-1+e^{-2}$. In general, you get

$$u(x)=x-1+e^{-\lfloor x \rfloor}$$

So on each part of the integration domain, $du=dx$ (constant doesn't make a difference).

However, it is quite unclear what exactly this integral means. It is not a proper Riemann integral in two ways.

First, it is ambiguous what the integral boundaries mean. Are they meant to refer to values of $u$ or $x$? If $x$, then it is quite unconventional notation, as $u$ to $x$ is not one to one (more $x$ have the same $u$).

The integral can be computed, as $du=dx$ almost everywhere, but you need to be careful, because technically, if $u(x)$ is treated as distribution, there is a discrete negative jump at $x=1$ and $x=2$ etc., so you have discrete (delta function) contributons in addition to the continuous integral (derivative of a Heaviside jump function is a delta function):

$${\rm d}u={\rm d}x(1+(e^{-1}-1)\delta(x-1)+\cdots)$$ Fortunately your integral does not encompass any of the jumps so the whole transformation is pointless, and you can simply ignore it:

$$\int_0^{\pi/4} f(x){\rm d}u=\int_0^{\pi/4} f(x){\rm d}x$$

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  • $\begingroup$ The limits refer to values of $x$. $\endgroup$ Jan 14, 2019 at 10:50
  • $\begingroup$ @Ashok Thank you, I modified my response. $\endgroup$
    – orion
    Jan 14, 2019 at 11:04
  • $\begingroup$ but since the integral has been restricted to $0 \to \frac{\pi}{4}$, wouldn't $x \not\ge 1$, hence dropping the discrete contributions? $\endgroup$ Jan 14, 2019 at 18:49
  • $\begingroup$ @Ashok you are right! That's embarassing. I completely missed it, in my head I was constantly thinking of $\pi/2$. $\endgroup$
    – orion
    Jan 14, 2019 at 19:04

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