A function of two variables

We have that $\displaystyle f(m,n) = \int_0^{m^n} \int_{\sqrt[n]{y}}^m e^{x^{n+1}} dxdy$ .

Here, we can set $\displaystyle y = x^n \implies \int_0^m \int_0^{x^n} e^{x^{n+1}} dydx \\ \displaystyle = \int_0^m \left[ y \cdot e^{x^{n+1}} \right]_0^{x^n} dx = \int_0^m x^n e^{x^{n+1}} dx \\ \displaystyle = \left[ \dfrac{1}{n+1} e^{x^{n+1}} \right]_0^m = \dfrac{1}{n+1} \left( e^{m^{n+1}} - 1 \right)$ .

Hence, $f(5,6) = \dfrac{1}{7} \left( e^{5^7} - 1 \right) \implies a+b+c+d = \boxed{20}$ .