This function consists of a cosine term, a cubic term, and a square root term. The cubic term is odd whereas the cosine and the square root term are even. If we distribute the parentheses, the integrand becomes the sum of a square root term and a cubic-cosine-square root term. The cubic-cosine-square root term is odd, so the integral from -2 to +2 is just zero.
Now we're left with the square root term. That's a half circle with radius 2 and centered on the origin. It's got a factor of 1/2 attached to it, so it's really a quarter circle of radius 2.
That area is pi, thus the answer is the first 10 digits of pi or 3.141592653