Find $\lim_{k \to \infty}\int_{0}^{\infty}ke^{-kx^2}\arctan(x)dx$

Find $\lim_{k \to \infty}\int_{0}^{\infty}ke^{-kx^2}\arctan(x)dx$.

I think that the limit is infinity.

$ke^{-kx^2}\leq ke^{-kx^2}\arctan(x)$ for $[\tan(1),\infty)$, but by integrating we know that $\int_{0}^{\infty}ke^{-kx^2}\to \infty$ and so out sequence of original integrals diveres too.

Is this correct?

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