October 23rd, 2018, 12:16 PM  #1 
Senior Member Joined: Jan 2012 Posts: 123 Thanks: 2  Plz help solve this integral...
How can we solve the following integral: $\displaystyle \int \frac{x^{3}1}{(x^{4}+1)(x+1)} dx$ Thx. 
October 23rd, 2018, 12:31 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,317 Thanks: 1230 
partial fractions expansion immediately suggests itself $\large \begin{align*} &\frac{x^31}{(x^4+1)(x+1)} = \frac{x^3}{x^4+1}\frac{1}{x+1}\\ \\ \displaystyle &\int~\frac{x^31}{(x^4+1)(x+1)}~dx = \\ \\ &\int ~ \frac{x^3}{x^4+1}\frac{1}{x+1}~dx =\\ \\ &u=x^4+1,~du=4x^3\\ \\ &\dfrac 1 4 \int \dfrac{du}{u}  \ln(x+1) + C = \\ \\ &\dfrac 1 4 \ln(x^4+1)  \ln(x+1)+C \end{align*}$ 
October 23rd, 2018, 12:51 PM  #3 
Senior Member Joined: Jan 2012 Posts: 123 Thanks: 2  Thx. Is there any way to do partial fraction of this expression where power 4 in denominator is there..or just observation.

October 23rd, 2018, 02:07 PM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 2,317 Thanks: 1230  
October 23rd, 2018, 06:36 PM  #5 
Senior Member Joined: Jan 2012 Posts: 123 Thanks: 2 
I mean for partial fractions, we have some basic methods. But in this question, partial fraction was done by observation or by method (like assuming A, B, etc..in the numerator and finding the constants, etc..)?

October 23rd, 2018, 07:09 PM  #6  
Senior Member Joined: Sep 2015 From: USA Posts: 2,317 Thanks: 1230  Quote:
 
October 23rd, 2018, 09:00 PM  #7 
Senior Member Joined: Jan 2012 Posts: 123 Thanks: 2 
Thanks for that table but only concern was about degree four polynomial in denominator which cannot be further factorized ($\displaystyle x^{4}+1$ in this question), for this particular factor, I guess we have to do partial fraction by observation.
Last edited by happy21; October 23rd, 2018 at 09:06 PM. 
October 23rd, 2018, 09:57 PM  #8  
Senior Member Joined: Sep 2015 From: USA Posts: 2,317 Thanks: 1230  Quote:
$\dfrac{x^31}{(x^4+1)(x+1)} = \dfrac{A x^3 + B x^3 + C x + D}{x^4+1}+\dfrac{E}{x+1}$  
October 23rd, 2018, 10:13 PM  #9 
Global Moderator Joined: Dec 2006 Posts: 20,303 Thanks: 1974  
October 24th, 2018, 05:05 AM  #10 
Senior Member Joined: Sep 2016 From: USA Posts: 559 Thanks: 324 Math Focus: Dynamical systems, analytic function theory, numerics  To expand on what skipjack has done, EVERY polynomial of degree $d > 2$ can be factored into lower degree polynomials over the reals. Also EVERY polynomial of degree $d > 1$ can be factored into lower degree polynomials over the complex numbers. Hence, partial fractions works universally for problems like this.


Tags 
integral, plz, solve 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How can I solve such an Integral?  ghafarimahsa  Calculus  1  September 11th, 2013 01:07 PM 
How to solve this integral?  kaspersky0  Complex Analysis  3  October 15th, 2012 10:19 AM 
How to solve an integral.  Ad van der ven  Calculus  3  December 17th, 2011 10:53 AM 
An integral to solve  malaguena  Calculus  0  February 14th, 2011 02:18 AM 
please solve this integral ..thx  art2  Calculus  1  September 24th, 2009 05:46 AM 