一类含对数的二重积分

青青子衿

\[ \int_0^1\int_0^1 \ln\left(x^\overset{\,}{2}+y^2\right){\rm\,d}x{\rm\,d}y=-\left(3-\frac{\pi}{2}-\ln2\right) \]
\[ \int_0^1\int_0^1 \left|\ln\left(x^\overset{\,}{2}+y^2\right)\right|{\rm\,d}x{\rm\,d}y=\pi-3+\ln2 \]

青青子衿

\[\int_0^1\int_0^1 \ln\left(x^\overset{\,}{2}+y^2\right){\rm\,d}x{\rm\,d}y=-\left(3-\frac{\pi}{2}-\ln2\right)\]
\[\int_0^1\int_0^1 \left|\ln\left(x^\overset{\,}{2}+y^2\right)\right|{\rm\,d}x{\rm\,d}y=\pi-3+\ln2\]
青青子衿 发表于 2018-12-13 18:12

\[ \int_0^1\int_0^1 \ln\left(x^\overset{\,}{3}+y^3\right)\mathrm{d}x\mathrm{d}y=-\left(\frac{9}{2}-\frac{\,\pi}{\sqrt{\,3\,}}-2\ln2\right) \]
\[ \int_0^1\int_0^1 \ln\left(x^\overset{\,}{4}+y^4\right)\mathrm{d}x\mathrm{d}y=-\left(6-\frac{\,\pi}{\sqrt{\,2\,}}-\sqrt{\,2\,}\ln\big(1+\sqrt{\,2\,}\,\big)-\ln2\right) \]