i try solve this improper integral $$\int_0^\infty x^p\sin x^q \ dx$$ I try to compare it with $\displaystyle \int_0^\infty\ \frac{1}{x^p}\ dx$

But I don't know what do when $x\rightarrow\ \infty$ in $\sin x^q$

Any hint I will really appreciate it.

Thanks for your corrections.

  • $\begingroup$ If all you need to do is show that the integral exists then remember that $|\sin t|\leq 1$. That won't cover all cases, but it is a start. $\endgroup$ – B. S. Thomson Oct 29 '15 at 23:57
  • $\begingroup$ Make a substitution $x^q=u$. $\endgroup$ – A.S. Oct 30 '15 at 0:40
  • $\begingroup$ Near $0$, use $\sin t\simeq t$. Near $\infty$, see the Riemann-Lebesgue lemma. $\endgroup$ – Lucian Oct 30 '15 at 4:53
  • $\begingroup$ I have $\int_0^\infty\ u^\frac{p-q+1}q\ sin\ u\ du$ $\endgroup$ – Camilo Acevedo. Oct 30 '15 at 4:54
  • $\begingroup$ Then you have convergence as long as the power $u$ is raised to is less then $0$ (for convergence around $\infty$) and greater then $-2$ (for convergence around $0$). $\endgroup$ – A.S. Oct 30 '15 at 11:57

If we set $x=z^{\frac{1}{q}}$ we have: $$ I(p,q) = \int_{0}^{+\infty}x^p\sin(x^q)\,dx = \frac{1}{q}\int_{0}^{+\infty}z^{\frac{p+1}{q}}\frac{\sin z}{z}\,dz $$ but: $$ \mathcal{L}\left(\frac{\sin z}{z}\right)=\arctan\frac{1}{s},\qquad \mathcal{L}^{-1}(z^{\alpha}) = \frac{s^{-1-\alpha}}{\Gamma(-\alpha)}$$ hence: $$ I(p,q)=\frac{1}{q\,\Gamma\left(-\frac{p+1}{q}\right)}\int_{0}^{+\infty}s^{-1-\frac{p+1}{q}}\arctan\left(\frac{1}{s}\right)\,ds $$ or: $$ I(p,q)=\frac{1}{q\,\Gamma\left(-\frac{p+1}{q}\right)}\int_{0}^{+\infty}t^{\frac{p+1}{q}}\frac{\arctan t}{t}\,dt $$ so:

$$ I(p,q) = -\frac{\pi}{2(p+1)\,\Gamma\left(-\frac{p+1}{q}\right)\cos\left(\frac{\pi(p+1)}{2q}\right)}$$

provided that $-1<\frac{p+1}{q}<0$.

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  • $\begingroup$ Why do you need $\frac {p+1}q<0$? Only $<1$ is needed for convergence. $\endgroup$ – A.S. Oct 30 '15 at 12:05
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    $\begingroup$ @A.S.: we have to grant integrability in a right neighbourhood of the origin and in a left neighbourhood of $+\infty$. $\endgroup$ – Jack D'Aurizio Oct 30 '15 at 12:59
  • $\begingroup$ $\sin$ is oscillating so the integral will converge conditionally for $\frac {p+1}q\in [0,1)$. $\endgroup$ – A.S. Oct 30 '15 at 13:04
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    $\begingroup$ @A.S.: oh, you're right, convergence is granted by $\left|\frac{p+1}{q}\right|<1$, but my manipulations through the Laplace transform are working only for a smaller range. I believe we may enlarge that range by integrating by parts, as usual. $\endgroup$ – Jack D'Aurizio Oct 30 '15 at 13:09

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