\(I = \int\limits _{x= 0}^a \int \limits_{y = 0}^x \int \limits_{z = 0}^{x + y} e^{x + y}. e^z \,dz\, dy\,dx\)
\(= \int\limits _{x= 0}^a \int \limits_{y = 0}^x e^{x + y}.[e^z]_0^{x + y}\, dy\,dx\)
\(= \int\limits _{x= 0}^a \int \limits_{y = 0}^x e^{x + y}(e^{x + y} - 1)\, dy\,dx\)
\(= \int\limits _{x= 0}^a \int \limits_{y = 0}^x (e^{2x}.e^{2y}- e^x . e^y)\, dy\,dx\)
\(= \int \limits_{x = 0}^a \left\{e^{2x}\left[\frac{e^{2y}}2\right]_0^x - e^x[e^y]_0^x\right\}dx\)
\(= \int \limits_{x = 0}^a \left\{\frac{e^{2x}}{2}(e^{2x}-1) - e^x(e^x - 1)\right\}dx\)
\(= \int \limits_{x = 0}^a \left(\frac{e^{4x}}2 - \frac 32 e^{2x}+ e^x\right)dx\)
\(= \left[\frac{e^{4x}}8 - \frac{3e^{2x}}4 + e^x\right]_0^a\)
\(= \left(\frac{e^{4a}}{8} - \frac{3e^{2a}}4 + e^a\right)-\left(\frac 18 - \frac34 + 1\right)\)
\(= \frac{e^{4a}}8 - \frac{3e^{2a}}4 + e^a - \frac 38\)
\(= \frac 18(e^{4a} - 6e^{2a} + 8e^{a} - 3)\)
