Integral X4 Cos X

Páginas: 2 (348 palabras) Publicado: 23 de abril de 2012
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 partssimplifies, 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 theintegration. 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 shouldconsider assigning them the role of v � .) We integrate by parts using the following assignments: u = x4 u� = 4x3 to get: � x4 cos x dx = x4 sin x − � � 4x3 sin x dx. v = sin x v �= cos x

We do not have a formula for parts will get us closer to one:

4x3 sin x dx, but a similar integration by



v = − cos x v � = sin x � � 3 3 4x sin x dx = −4xcos x + 12x2 cos x dx.

u = 4x3 u� = 12x2

We must integrate by parts twice more before we can finish the problem. u = 12x2 u� = 24x � ⇒
2

v = sin x v � = cos x
2

� 24xsin x dx.

12x cos x dx = 12x sin x − u = 24x u� = 24 � 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 assemblethem 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 + 24 cos x dx)

� 24x sin x dx)

= x4 sin x + 4x3 cos x − 12x2 sin x − 24x cos x + 24 sin x
+c.


2

MIT OpenCourseWarehttp://ocw.mit.edu

18.01SC Single Variable Calculus��
Fall 2010 ��

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

Leer documento completo

Regístrate para leer el documento completo.

Estos documentos también te pueden resultar útiles

  • X cosa
  • X Cosas
  • Cosas x grasa
  • el uso de x cosa en X cosa
  • X Cosa
  • x cosa
  • X cosa
  • Cosas 'x'

Conviértase en miembro formal de Buenas Tareas

INSCRÍBETE - ES GRATIS