Am I allowed to change limit and integral here? How can I solve this integral?
$$\lim_{a\rightarrow 0} \int_0^1 \frac{1}{1+a\sin(x)} \, \mathrm{d}x$$
2026-04-11 18:34:44.1775932484
$\lim_{a\rightarrow 0} \int_0^1 \frac{1}{1+a\sin(x)} \, \mathrm{d}x$
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For $x\in[0,1]$ and $|a|<1/2$, \begin{align*} \left|\dfrac{1}{1+a\sin x}\right|\leq\dfrac{1}{1-(1/2)\sin(1)} \end{align*} and \begin{align*} \int_{0}^{1}\dfrac{1}{1-(1/2)\sin 1}dx<\infty, \end{align*} so it goes through by Lebesgue Dominated Convergence Theorem.