integrate sqrt(2((sin(x))^2 + 50((cos(x))^2 - 10sin(x)cos(x))
You sure you don't mean 20sinx*cosx? That would give
√(2sin^2(x) + 50cos^2(x) - 20sinx*cosx)
√2 √(sin^2x + 10sinx*cosx + 25cos^2x)
√2 √(sinx+5cosx)^2
√2 (sinx + cosx)
which would be easy to integrate.
As it is given, there is no integral using elementary functions.