Example 4

Find . Let u = 2 - x², du = - 2x dx, dx = du/( - 2x). We try to express the integral in terms of u.

Since u = 2 - x², x² = 2 - u. Therefore

We next describe the method of definite integration by change of variables. In a definite integral

it is always understood that x is the independent variable and we are integrating between the limits x = a and x = b. Thus when we change to a new independent variable u, we must also change the limits of integration. The theorem below will show that if u = c when x = a and u = d when x = b, then c and d will be the new limits of integration.