Analysis:
To find the integral values of n for which the equation sinx(2sinx + cosx) = n has at least one real solution, we need to analyze the given equation and understand its behavior.

Understanding the Equation:
The equation sinx(2sinx + cosx) = n can be simplified as 2sin^2(x) + cos(x)sin(x) = n.
Further simplifying, we get 2sin^2(x) + sin(2x)/2 = n.

Analysis of Solutions:
For the equation to have at least one real solution, the discriminant of the quadratic equation should be non-negative.
The discriminant of 2sin^2(x) + sin(2x)/2 - n = 0 is given by (sin(2x)/2)^2 - 4*2*(-n) >= 0.
Solving this inequality will give us the range of values of n for which the equation has real solutions.

Finding Integral Values of n:
To find the integral values of n, we need to solve the inequality and determine the range of n values that satisfy the condition for real solutions.
By finding the values of n that satisfy the inequality, we can sum up the integral values of n that meet the criteria.

Conclusion:
By analyzing the given equation, understanding its behavior, and solving the inequality for the discriminant, we can determine the integral values of n for which the equation sinx(2sinx + cosx) = n has at least one real solution.