Show that $ \int_I x^{-\frac{1}{4}} $sin$ x \;dx \leq \pi^{\frac{3}{4}}$.

Question

Let $I=[0,\pi]$. Show that $\displaystyle \int_I x^{-\frac{1}{4}} $sin$ x \;dx \leq \pi^{\frac{3}{4}}$.

My Work:

I think this is an application of Holders inequality. But any positive power of $ x^{-\frac{1}{4}}$ is not integrable over $I$. So I was stuck. Can anyone please give me a hint for the start.

Answer 1

You can use Hölder's inequality like this: $$\int_I x^{-1/4}\sin x\,{\rm d}x \leq \left(\int_I (x^{-1/4})^{4/3}\,{\rm d}x\right)^{3/4}\left(\int_I\sin^4 x\, {\rm d}x\right)^{1/4} \leq \left(\int_I x^{-1/3}\,{\rm d}x\right)^{3/4}.$$