Jessie Lee

2022-01-17

Prove that $\underset{t\to \mathrm{\infty }}{lim}{\int }_{1}^{t}\mathrm{sin}\left(x\right)\mathrm{sin}\left({x}^{2}\right)dx$ converges

RizerMix

You can write
$\mathrm{sin}\left(x\right)\mathrm{sin}\left({x}^{2}\right)=\frac{1}{2}\left[\mathrm{cos}\left({x}^{2}-x\right)-\mathrm{cos}\left({x}^{2}+x\right)\right]$and let $u={x}^{2}-x,v={x}^{2}+x$ to obtain

The convergence of this expression as $t\to \infty$ is ensured by Dirichlet's test for integrals or integration by parts.

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