integral of (1+sqrt(x))^8

It looks much worse than it really is. It is actually very easy. Just do a simple u substitution for the square root of x, and then manipulate the equation du/dx by substituting back in a u in place of the radical.

The u-substitution of the square-root of x is such that the u can be substituted back again in the equation of du/dx. This is because the derivative of square-root of x is 2(square-root of x)^(-1). This allows us to say that dx is equal to 2udu, and we can be rid of the radical until the end of the problem.