In calculating a number of integrals we had to use the method of integration by parts several times in succession. The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts

u(x) v(x) dx = u(x) v 1 (x) – u ′ (x) v 2 (x) + u ′′ (x) v 3 (x) + ......+ (–1) n–1 u n–1 (x) v n (x) –

(–1) n–1 u n (x) v n (x) dx Where v 1 (x) = v(x) dx, v 2 (x) = v 1 (x) dx ..,v n (x) = v n–1 (x) dx

Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when calculating P n (x), Q(x)dx, where P n (x) is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n + 1 times

(i) If (x 3 –2x 2 + 3x –1) cos 2x dx = u(x)

+ v(x) + C then