Cubic functions and Trigometric functions

$f(0) = g(0) = 1 \implies d = 1$ and $f(\dfrac{\pi}{2}) = g(\dfrac{\pi}{2}) = 1 \implies \boxed{2\pi b + 4c = -\pi^2}$ and $\int_{0}^{\dfrac{\pi}{2}} g(x) dx$ = $\int_{0}^{\dfrac{\pi}{2}} \sin(x) + \cos(x) dx = \sin(x) - \cos(x)|_{0}^{\dfrac{\pi}{2}} = 2 \implies$

$2 = \int_{0}^{\dfrac{\pi}{2}} (x^3 + bx^2 + cx + 1) dx = \dfrac{1}{4}x^4 + \dfrac{b}{3}x^3 + \dfrac{c}{2}x^2 + x|_{0}^{\dfrac{\pi}{2}} = \dfrac{\pi^4}{2^6} + \dfrac{\pi^3}{3 * 2^3}b + \dfrac{\pi^2}{2^3}c + \dfrac{\pi}{2} \implies$

$\boxed{8\pi^3b + 24\pi^2c = 384 - 96\pi - 3\pi^4}$

$\boxed{2\pi b + 4c = -\pi^2}$

Solving the above system we obtain:

$c = \dfrac{384 - 96\pi + \pi^4}{8\pi^2}$ and $b = -\dfrac{384 - 96\pi + 3\pi^4}{4\pi^3} \implies$

$\int_{\dfrac{-\pi}{4}}^{0} f(x) dx = \dfrac{x^4}{4} - \dfrac{384 - 96\pi + 3\pi^4}{3 * 4\pi^3} x^3 + \dfrac{384 - 96\pi + \pi^4}{2 * 8\pi^2} x^2 +x|_{\dfrac{-\pi}{4}}^{0} =$ $-\dfrac{27\pi^4 - 2304\pi + 6144}{3072}$

$\int_{\dfrac{-\pi}{4}}^{0} g(x) dx = \int_{\dfrac{-\pi}{4}}^{0} \sin(x) + \cos(x) dx = \sin(x) - \cos(x)|_{\dfrac{-\pi}{4}}^{0} = \sqrt{2} - 1$

$\implies \int_{\dfrac{-\pi}{4}}^{0} g(x) - f(x) dx = \dfrac{9\pi^4 - 768\pi + 1024(\sqrt{2} + 1)}{1024} =$

$\sqrt{2} + 1 - \dfrac{3\pi}{4} + \dfrac{9\pi^4}{1024} = \sqrt{2} + 1 - \dfrac{3\pi}{2^2} + (\dfrac{3\pi^2}{2^5})^2 =\sqrt{\alpha} + \beta - \dfrac{\lambda\pi}{\alpha^{\alpha}} + (\dfrac{\lambda\pi^{\alpha}}{\alpha^{\gamma}})^{\alpha} \implies \alpha + \beta + \lambda + \gamma = \boxed{11}$ .