Prove that \lim_{t \to \infty} \int_1^t \sin(x) \sin(x^2)dx converges

Jessie Lee

Jessie Lee

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Prove that limt1tsin(x)sin(x2)dx converges

Answer & Explanation



Expert2022-01-19Added 656 answers

You can write
sin(x)sin(x2)=12[cos(x2x)cos(x2+x)]and let u=x2x,v=x2+x to obtain
1tsin(x)sin(x2)dx=121t[cos(x2x)cos(x2+x)] =14[0t2tcos(u)u+14du2t2+tcos(v)v+14dv]
The convergence of this expression as t is ensured by Dirichlet's test for integrals or integration by parts.

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