数学题在线解答

问题:瑕积分:设a>0, 则∫lnx/x²+a²dx=?

瑕积分

解答

\begin{aligned}
&\int_{0}^{+\infty} \frac{\ln x}{x^{2}+a^{2}} d x\\
=&\int_{0}^{+\infty} \frac{ln (a u)}{(a u)^{2}+a^{2}} \operatorname{dau}(x=a u)\\
=& \frac{1}{a} \int_{0}^{+\infty} \frac{\ln (a)+ln (u)}{u^{2}+1} d u\\
=&\frac{\ln (a)}{a} \int_{0}^{+\infty} \frac{1}{u^{2}+1} d u+\frac{1}{a} \int_{0}^{+\infty} \frac{\ln (u)}{u^{2}+1} du\
=&\left.\frac{\ln (a)}{a} \arctan (u)\right|_{0} ^{\infty}+\frac{1}{a} \cdot 0\\
=& \frac{\ln (a) \cdot \pi}{2 a}
\end{aligned}

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