Integral of uⁿu'dx with a factor x missing:

This looks like it has the form of a quantity raised to a power, handled by the General Power rule for integrals, but we shall find that it doesn't and will have to use another method. To see this,
let , n = 2, and . There's the problem: we have the necessary -10 present in the original problem, but not the factor x. Consequently, we can't use this rule. Since the quantity is only squared, we can do the multiplication and see if that result can be handled by any of our rules. We get

This is a polynomial, so we can use the Sum rule for integrals to get

Now use the Scalar Multiple rule for integrals to get

Finally, we use the Power rule for integrals to get

Test Problem: Find

a)

b)

c)

d)