Use integration by parts to find the indefinite integral for each of the following functions involving a product. Check your answers on your own, recalling that the product rule is needed to check an answer obtained through integration by parts.

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(a) (b) - x(x-8) dx Picking the simpler function for f(x) and the more complicated function for g'(x), let f(x) = x and g'(x) = (x8). Then f'(x) = 1 and g(x) = f(x 8) dx = (x-8)4 + c, where c will drop out and can be ignored, as explained in Example 10. Now substituting the relevant values in (12.6), - || f(x) - x(x 8) dx = f (x) g(x) [g(x) f'(x)] dx =X- 1 - - [[8(x)- - {(x 8) / [{ }(x 8) - (1)] d. 8)4. dx x = - = (x 8) - (x 8)* dx - - Then substituting (x 8) 4 dx = 20 (x 8)5 + c above, - - x(x = 8) dx = (x 8) - x - (-8)5+ +c 4 20 56x(x+11) dx Letting f(x) = 56x and g'(x) = (x+11)6, then f'(x) = 56 and g(x) = = (x+11). Substituting in (12.6), 1 11)7.56 dx 56x - = (x + 11)? - [ [ (x+11)? - [ 56x(x+11) dx = = 8x (x + 11)7 - 8 8(x+11) dx Integrating once again, Js 56x(x+11) dx = 8x(x + 11)7 - (x+11) 8+c