We choose *u* = (*x*^{2} + 4). Then *u*' = 2*x*. We have the factor *x* in the integrand, but we're missing the 2. It's time for a trick. Multiply the integrand by Since , we're not changing the value of the integrand. Now we do have 2*x* as a factor of the integrand, so we can change variables: Using one of the properties of integrals we can pull the constant out of the integral. Then we can integrate and put the original variable back in: As our final answer we write You might reasonably wonder how the turned into just + *C*. The expression accounts for the entire family of antiderivatives of (*x*^{2} + 4)^{7}*x*. So does the expression Since both these expressions describe the same collection of functions, it doesn't matter which one we use. So we might as well use the simpler one and just write + *C*. |