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一类较复杂的三重积分

本帖最后由 青青子衿 于 2018-11-12 18:51 编辑

\[ \int_0^c\int_0^b\int_0^a\left(x^2+y^2\right)\sqrt{x^{\overset{\,}{2}}+y^2+z^2}{\rm\,d}x{\rm\,d}y{\rm\,d}z \]
\begin{align*}
\color{red}{\frac{abc\left(43a^2+43b^2+10c^2\right)}{360}\sqrt{a^{\overset{\,}{2}}+b^2+c^2}+\frac{c^6}{45}\arctan\left(\frac{ab}{c\sqrt{a^2+b^2+c^2}}\right)}\\
\color{red}{-\frac{2a^6}{45}\arctan\left(\frac{bc}{a\sqrt{a^2+b^2+c^2}}\right)-\frac{2b^6}{45}\arctan\left(\frac{ac}{b\sqrt{a^2+b^2+c^2}}\right)}\\
\color{red}{+\frac{ac\left(27a^4+10a^2c^2-9c^4\right)}{360}\operatorname{arsinh}\left(\frac{b}{\sqrt{a^2+c^2}}\right)}\\
\color{red}{+\frac{bc\left(27b^4+10b^2c^2-9c^4\right)}{360}\operatorname{arsinh}\left(\frac{a}{\sqrt{b^2+c^2}}\right)}\\
\color{red}{+\frac{ab\left(9a^4+10a^2b^2+9b^4\right)}{90}\operatorname{arsinh}\left(\frac{c}{\sqrt{a^2+b^2}}\right)}\\
\end{align*}
\begin{align*}
\frac{abc\left(43a^2+43b^2+10c^2\right)}{360}\sqrt{a^{\overset{\,}{2}}+b^2+c^2}+\frac{c^6}{45}\arctan\left(\frac{ab}{c\sqrt{a^2+b^2+c^2}}\right)\\
-\frac{2a^6}{45}\arctan\left(\frac{bc}{a\sqrt{a^2+b^2+c^2}}\right)-\frac{2b^6}{45}\arctan\left(\frac{ac}{b\sqrt{a^2+b^2+c^2}}\right)\\
+\frac{ac\left(27a^4+10a^2c^2-9c^4\right)}{360}\ln\left(\frac{b}{\sqrt{a^2+c^2}}+\sqrt{1+\frac{b^2}{a^2+c^2}}\,\right)\\
+\frac{bc\left(27b^4+10b^2c^2-9c^4\right)}{360}\ln\left(\frac{a}{\sqrt{b^2+c^2}}+\sqrt{\frac{a^2}{b^2+c^2}+1}\,\right)\\
+\frac{ab\left(9a^4+10a^2b^2+9b^4\right)}{90}\ln\left(\frac{c}{\sqrt{a^2+b^2}}+\sqrt{1+\frac{c^2}{a^2+b^2}}\,\right)\\
\end{align*}
\[ \iiint\limits_{x^\overset{\,}{2}+y^2+z^2\le a^2} \left(x^2+y^2\right)\sqrt{x^2+y^2+z^2}{\rm\,d}x{\rm\,d}y{\rm\,d}z \]
\[ \dfrac{4\pi}{9}a^6 \]

\begin{align*}
\color{red}{\frac{abc\left(5a^2+5b^2+38c^2\right)}{360}\sqrt{a^{\overset{\,}{2}}+b^2+c^2}-\frac{c^6}{18}\arctan\left(\frac{ab}{c\sqrt{a^2+b^2+c^2}}\right)}\\
\color{red}{+\frac{a^6}{90}\arctan\left(\frac{bc}{a\sqrt{a^2+b^2+c^2}}\right)+\frac{b^6}{90}\arctan\left(\frac{ac}{b\sqrt{a^2+b^2+c^2}}\right)}\\
\color{red}{+\frac{ac^3\left(5a^2+9c^2\right)}{90}\operatorname{arsinh}\left(\frac{b}{\sqrt{a^2+c^2}}\right)}\\
\color{red}{+\frac{bc^3\left(5b^2+9c^2\right)}{90}\operatorname{arsinh}\left(\frac{a}{\sqrt{b^2+c^2}}\right)}\\
\color{red}{-\frac{ab\left(9a^4+10a^2b^2+9b^4\right)}{360}\operatorname{arsinh}\left(\frac{c}{\sqrt{a^2+b^2}}\right)}\\
\end{align*}
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