1∫0x (x − 1)(2c1 − 10x 2 − 5) dx =

The integral you provided is a definite integral from 0 to 1 of the function x(x - 1)(2c1 - 10x^2 - 5). To solve this, we can use the method of integration by parts, which is derived from the product rule of differentiation. The formula for integration by parts is ∫udv = uv - ∫vdu.

However, in this case, the integral simplifies greatly due to the limits of integration. Notice that the function x(x - 1) equals zero at both x = 0 and x = 1. Because of this, the integral from 0 to 1 of this function times any other function will also be zero, since we're multiplying by zero at both endpoints.

So, without doing any further calculations, we can say that the integral from 0 to 1 of x(x - 1)(2c1 - 10x^2 - 5) dx equals zero.