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How to solve $\left(\int_0^x t\sin(\frac{1}{t})dt\right)’_{x=0}$?

I’m not sure if I can use Newton-Leibniz Theorem $$\dfrac{d}{dx}\left(\int_a^xf(t) dt=f(x)\right)$$ because $f(x)=x\sin(\frac{1}{x})$ is undefined where $x=0$. Is it correct that let $f(0)=0$?

Robert Z
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闫嘉琦
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    What are the conditions required to apply the theorem ? –  Nov 20 '17 at 09:21
  • The problem becomes more interesting if $t\sin(1/t)$ is replaced by $\sin(1/t)$ or $\cos(1/t)$. In this case the discontinuity at $0$ can not be removed and it requires some more manipulation to get the answer as $0$. – Paramanand Singh Nov 21 '17 at 03:34

3 Answers3

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The fundamental theorem of calculus only requires your function to be continuous over a closed domain ; in our case $f : t \mapsto t \sin(1 / t)$ is continuous on $(0, a]$ for any $a$, so we can instead use the continuous continuation $\widetilde{f}$ which is equal to $f$ over $\mathbb{R}^*_+$ and is $0$ for $x = 0$ (this is indeed the limit of $f$, as $\sin$ is bounded by $1$). It turns out that $f$ and $\widetilde{f}$ have the same integral over any closed interval of $\mathbb{R}^*_+$ ; can you take it from there ?

Matrefeytontias
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  • I see. But use the fundamental theorem I get the $F’(x)=x\sin{\dfrac{1}{x}}$. To get $F’(0)$, can I let $x\to 0$, directly? – 闫嘉琦 Nov 20 '17 at 09:43
  • Absolutely, you can just do that. – Matrefeytontias Nov 20 '17 at 09:46
  • Because $f$ is continuous? Thanks a lot. – 闫嘉琦 Nov 20 '17 at 09:47
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    As I said, $f$ is not continuous over a closed interval because it is not defined at $0$. Instead, we consider the continuation, which is in turn defined at $0$ and stays continuous. Since this continuation is continuous and only differs from $f$ at one single point, both integrals are equal and you can apply the fundamental theorem of calculus to the continuation, leading to your answer. – Matrefeytontias Nov 20 '17 at 09:49
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Hint. Let $F(x):=\int_0^x t\sin(1/t)dt$, then $F(0)=0$ (note that $\lim_{t\to 0^+} t\sin(1/t)=0$ which means that the integrand function is bounded in a neighbourhood of $0$). Hence $F$ is a continuous function and $$\left(\int_0^x t\sin(1/t)dt\right)’_{x=0}=\lim_{x\to 0}\frac{F(x)-F(0)}{x-0}=\lim_{x\to 0}\frac{1}{x}\int_0^x t\sin(1/t)dt.$$ Now we may use L'Hopital rule (and Newton-Leibniz Theorem for $x\not=0$). You may also use the Mean Value Theorem: for any $x\not=0$ there is $t_x$ between $0$ and $x$ such that $$\frac{1}{x}\int_0^x t\sin(1/t)=t_x\sin(1/t_x).$$

P.S. More generally if a derivative of a continuous function has a limit, it must agree with that limit. See prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$.

Robert Z
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Let , $$f(x) = \int_0^x t\sin(1/t)dt$$

$$\left(\int_0^x t\sin(1/t)dt\right)’_{x=0}=f'(0)=\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}=\lim_{x\to 0}\frac{1}{x}\int_0^x t\sin(1/t)dt.$$

But since, $|\sin(1/t)|\le 1$

$$\left|\lim_{x\to 0}\frac{1}{x}\int_0^x t\sin(1/t)dt\right|\le \lim_{x\to 0}\frac{1}{x}\int_0^x tdt = \lim_{x\to 0}\frac{x}2 =0$$

Thus, $$f'(0)=\left(\int_0^x t\sin(1/t)dt\right)’_{x=0} = 0$$

Also see here for similar problem: Compute $f'(0)$ and check if $f'$ is continuous or not

Guy Fsone
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