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变限积分

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PIG Posted at 2021-11-26 17:24:52 |Read mode
关于一道变限积分的题,答案是1/10
  1. $\lim_{x\to0+}\frac{\int_0^\sqrt{a}(1-\sin\sqrt{x})dx}{x^frac{5}{2}$
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求大佬解答

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 Author| PIG Posted at 2021-11-26 17:31:56
原题如图
211126173193ee8ee7c301ce79.jpg

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abababa Posted at 2021-11-26 18:06:55
回复 2# PIG

就直接写代码,不用放在[code]里面,就是不用点“代码”那个。

\begin{align*}
\lim_{x\to 0}\frac{1}{x^{5/2}}\cdot\int_{0}^{\sqrt{x}}(1-\cos t^2)dt
&\xlongequal{罗比达} \lim_{x\to 0}\frac{1}{\frac{5}{2}x^{3/2}}\cdot(1-\cos x)\cdot(\sqrt{x})'\\
&=\lim_{x\to 0}\frac{1}{5}\cdot\frac{1-\cos x}{x^2}\\
&\xlongequal{罗比达}\lim_{x\to 0}\frac{1}{5}\cdot\frac{\sin x}{2x}
=\frac{1}{10}
\end{align*}

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 Author| PIG Posted at 2021-11-26 18:53:25
回复 4# abababa


    谢谢大佬,已经懂了

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