Integral of x4 cos x This problem provides a lot of practice with integration by parts. Compute the integral of x4 cos x. Solution A single application of integration by parts simpliﬁes, but does not solve, this integral. We must repeat integration by parts several times (or look up a re duction formula for integrating xn cos x) to complete the integration. Taking the derivative of cos x does not simplify it, while the derivative of x4 is slightly simpler. (In general, if we see sine, cosine or exponential functions, we should consider assigning them the role of v � .) We integrate by parts using the following assignments: u = x4 u� = 4x3 v = sin x v � = cos x to get: � x4 cos x dx = x4 sin x − We do not have a formula for parts will get us closer to one: � � 4x3 sin x dx. 4x3 sin x dx, but a similar integration by u = 4x3 u� = 12x2 ⇒ v = − cos x v � = sin x � � 3 3 4x sin x dx = −4x cos x + 12x2 cos x dx. We must integrate by parts twice more before we can ﬁnish the problem. u = 12x2 u� = 24x � ⇒ v = sin x v � = cos x 2 2 12x cos x dx = 12x sin x − u = 24x u� = 24 24x sin x dx. v = − cos x v � = sin x � ⇒ � � 24x sin x dx = −24x cos x + 24 cos x dx. Now we have all the pieces and can assemble them into our ﬁnal answer: � � 4 4 x cos x dx = x sin x − 4x3 sin x dx � = x4 sin x − (−4x3 cos x + 12x2 cos x dx) 1 = x4 sin x + 4x3 cos x − (12x2 sin x − = x4 sin x + 4x3 cos� x − 12x2 sin x +(−24x cos x + = � 24x sin x dx) 24 cos x dx) x4 sin x + 4x3 cos x − 12x2 sin x − 24x cos x + 24 sin x +c. 2