Use integration by parts to find the indefinite integral for each of the following natural exponential functions.

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(a) (x+5)e (x+5)e* dx Let f(x)=x+5 and g'(x) = ex; then f'(x) = 1 and g(x) = fe* dx = e*. Substituting, (b) for+5) (x+5)e dx = (x+5)e* - fe dx ex Substituting ex dx = e* + c above, (x+5)e dx=(x+5)e-ex + c 24xe6x dx Let f(x)=24x2 and g'(x) = ex. Then f'(x) = 48x and g(x) = ex. Substituting in (12.6), 24xe6x dx = 24x. fex - [ex. 48x] dx = 4xex - 8xe 8xex dx Using integration by parts again, 24xex dx = 4xex - [8x fear - f (ex. 8) dx] - = 4xex - xe6x + ex dx = 4xex-xex + x + c 6x