Question

Calculate The Double Integral. ∫∫ 4x 1 + Xy DA, R = [0, 4] × [0, 1]

Answer 1

`20\ln(5)-16\approx16.18876`

Explanation:

`\displaystyle\int\int\limits^{}_R {(4x)/(1+xy) } \, dA\\\\\\=\int\limits^4_0\int\limits^1_0 {(4x)/(1+xy) } \, dy \, dx\\\\\\=\int\limits^4_0 \ln(1+xy)\biggr \, dx\\\\\\=\int\limits^4_0 {\ln(1+x)} \, dx\,\,\,\,\,\,\,\,\,\,\,\,\bigr(u=\ln(1+x),\,dv=dx\bigr)\\\\\\=4\biggr(x\ln(1+x)\biggr|_0^4-\int\limits^4_0 (x)/(1+x) \,dx\biggr)\\\\=4\biggr(4\ln(5)-\bigr[x-\ln(1+x)\bigr]\biggr|_0^4\biggr)\\\\=4\biggr(4\ln(5)-\bigr[4-\ln(5)\bigr]\biggr)\\\\=4\biggr(5\ln(5)-4]\biggr)\\\\=20\ln(5)-16`