JcJC l4 4ci J [x l4 4cJ dx

3. Use Rule 3 to compute the final integral and substitute.

4x(x + l)3 dx = x(x + l)4 + 4cxx - l(x + l)5 -4cxx + c = x(x + l)4~l(x + l)5 + c

Note that the cx term does not appear in the final solution. Since this is common to integration by parts, Ci will henceforth be assumed equal to 0 and not formally included in future problem solving.

4. Check the answer by letting = x(x + l)4 - \(x + l)5 + c and using the product and generalized power function rules.

/(*) = [x • 4{x + l)3 + (jc + l)4 • 1] - (x + l)4 = 4x{x + l)3

EXAMPLE 8. The integral f Ixe* dx is determined as follows:

Let f(x) = 2x and g'(x) = e?\ then f'(x) = 2, and by Rule 6, g(x) = Jexdx = ex. Substitute in (14.1).

Apply Rule 6 again and remember the constant of integration.

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