Let's do some calculus! (42)

$\begin{aligned} I & = \int_0^\frac \pi 2 \sum_{n=1}^5 \sin^n x \sum_{m=1}^5 \cos^m x \ dx \\ & = \int_0^\frac \pi 2 \sum_{n=1}^5 \sum_{m=1}^5 \sin^n x \cos^m x \ dx \\ & = \sum_{n=1}^5 \sum_{m=1}^5 \int_0^\frac \pi 2 \sin^n x \cos^m x \ dx \\ & = \frac 12 \sum_{n=1}^5 \sum_{m=1}^5 B \left(\frac {n+1}2, \frac {m+1}2 \right) & \small \color{#3D99F6} \text{where } B(m, n) \text{ is beta function.} \\ & = \frac 12 \sum_{n=1}^5 \sum_{m=1}^5 \frac {\Gamma \left(\frac {n+1}2 \right)\Gamma \left(\frac {m+1}2 \right)}{\Gamma \left(\frac {n+m}2 + 1 \right)} & \small \color{#3D99F6} \text{where } \Gamma (x) \text{ is gamma function.} \\ & = \frac 12 \left( \frac {91}{30} + \frac {208}{63} + \frac {35}{128}\pi \right) & \small \color{#3D99F6} \text{See Note.} \\ & = \frac {3991}{1260} + \frac {35}{256}\pi \end{aligned}$

$\implies a + b + c + d = 3991+1260+35+256 = \boxed{5542}$

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Note:

When both $m$ and $n$ are odd:

$\begin{aligned} S_{oo} & = B(1,1) + 2B(1,2) + 2B(1,3)+B(2,2) + 2B(2,3) + 2B(3,3) \\ & = \frac {0!0!}{1!} + 2 \times \frac {0!1!}{2!} + 2 \times \frac {0!2!}{3!} + \frac {1!1!}{3!} + 2 \times \frac {1!2!}{4!} + \frac {2!2!}{5!} \\ & = 1 + 1 + \frac 23 + \frac 16 + \frac 16 + \frac 1{30} \\ & = \frac {91}{30} \end{aligned}$

When either $m$ or $n$ is odd and the other even:

$\begin{aligned} S_{oe} & = 2 \left[ B \left(1, \frac 32\right) + B \left(1, \frac 52\right) + B \left(2, \frac 32\right) + B \left(2, \frac 52\right) + B \left(3, \frac 32\right) + B \left(3, \frac 52\right) \right] \\ & = 2 \left[\frac {0! \cdot \frac 12 \sqrt \pi}{\frac 32 \cdot \frac 12 \sqrt \pi} + \frac {0! \cdot \frac 32 \cdot \frac 12 \sqrt \pi}{\frac 52 \cdot \frac 32 \cdot \frac 12 \sqrt \pi} + \frac {1! \cdot \frac 12 \sqrt \pi}{\frac 52 \cdot \frac 32 \cdot \frac 12 \sqrt \pi} + \frac {1! \cdot \frac 32 \cdot \frac 12 \sqrt \pi}{\frac 72 \cdot \frac 52 \cdot \frac 32 \cdot \frac 12 \sqrt \pi} + \frac {2! \cdot \frac 12 \sqrt \pi}{\frac 72 \cdot \frac 52 \cdot \frac 32 \cdot \frac 12 \sqrt \pi} + \frac {2! \cdot \frac 32 \cdot \frac 12 \sqrt \pi}{\frac 92 \cdot \frac 72 \cdot \frac 52 \cdot \frac 32 \cdot \frac 12 \sqrt \pi} \right] \\ & = 2 \left[ \frac 23 + \frac 24 + \frac 4{15} + \frac 4{35} + \frac {16}{105} + \frac {16}{315} \right] \\ & = \frac {208}{63} \end{aligned}$

When both $m$ and $n$ are even:

$\begin{aligned} S_{ee} & = B \left(\frac 32,\frac 32\right) + 2B \left(\frac 32,\frac 52\right) + B \left(\frac 52,\frac 52\right) \\ & = \frac {\frac 12 \sqrt \pi \cdot \frac 12 \sqrt \pi}{2!} + 2 \times \frac {\frac 12 \sqrt \pi \cdot \frac 32 \cdot \frac 12 \sqrt \pi}{3!} + \frac {\frac 32 \cdot \frac 12 \sqrt \pi \cdot \frac 32 \cdot \frac 12 \sqrt \pi}{4!} \\ & = \frac 18 \pi + \frac 18 \pi + \frac 3{128} \pi \\ & = \frac {35}{128} \pi \end{aligned}$