Hitunglah \( \displaystyle \int_1^{\ln 8} \int_0^{\ln y} \ e^{x+y} \ dx \ dy \).
Pembahasan:
\begin{aligned} \int_1^{\ln 8} \int_0^{\ln y} \ e^{x+y} \ dx \ dy &= \int_1^{\ln 8} [e^{x+y}]_0^{\ln y} \ dy \\[8pt] &= \int_1^{\ln 8} (e^{\ln y+y}-e^y) \ dy \\[8pt] &= \int_1^{\ln 8} (ye^y-e^y) \ dy \\[8pt] &= \left[ ye^y-2e^y \right]_1^{\ln 8} = [e^y(y-2)]_1^{\ln 8} \\[8pt] &= e^{\ln 8} (\ln 8 - 2) - e (1 - 2) \\[8pt] &= 8(\ln 8 - 2)+e \\[8pt] &= 8 \ln 8 - 16 + e \end{aligned}
Catatan: untuk menyelesaikan integral \(\int \ ye^y \ dy\) kita bisa gunakan teknik integral parsial.