
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
January 19th, 2018, 03:31 AM  #1 
Member Joined: Sep 2014 From: Sweden Posts: 94 Thanks: 0  Integrals of type x^ncos(ax) or x^nsin(ax)?
How can you solve for these types of integrals? $\displaystyle \displaystyle \int x^{n}\cos(ax) \ dx \ or \displaystyle \int x^{n}\sin(ax) \ dx\\ $ An example of an integral that I want to solve $\displaystyle \displaystyle \int x^{5}\cos(3x) \ dx $ Using usubstitution: $\displaystyle \displaystyle u = \cos(3x)\\ \displaystyle du = 3\sin(3x) \ dx\\ $ Then going back to solve the integral $\displaystyle $\displaystyle \int x^{5}u \ du$ $ But how do I go on from here? 
January 19th, 2018, 04:20 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Use integration by parts, repeatedly. To integrate $\int x^5 \cos(x) dx$, let $u= x^5$ and $dv= \cos(x)dx$. Then $du= 5x^4 dx$ and $v= \sin(x)$. So the integral becomes $x^5\sin(x) 5\int x^4 \sin(x) dx$. Now let $u= x^4$ and $dv= \sin(x)$. Then $du= 4x^3 dx$ and $v= \cos(x)$. So the integral becomes $x^5\sin(x)+ 5x^4\cos(x) 20\int x^3 \cos(x)dx$. Keep doing that until the power of $x$ is reduced to 0. Last edited by skipjack; January 19th, 2018 at 04:25 AM. 
January 19th, 2018, 04:40 AM  #3 
Senior Member Joined: Oct 2009 Posts: 780 Thanks: 279 
I have nothing of substance further to add. But I think that you can simplify a lot of the computations if you instead compute $$\int x^n e^{iax}dx$$ 
January 19th, 2018, 06:30 AM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,934 Thanks: 1128 Math Focus: Elementary mathematics and beyond 
$$\cos(ax)=\frac{e^{iax}+e^{aix}}{2}$$ Substitute, expand and use a single usubstitution to evaluate. 
January 19th, 2018, 07:56 AM  #5 
Senior Member Joined: Sep 2016 From: USA Posts: 609 Thanks: 378 Math Focus: Dynamical systems, analytic function theory, numerics 
Personally, I would expand as a power series and integrate term by term. However, the other methods presented here are just fine as well.

January 19th, 2018, 09:08 AM  #6 
Math Team Joined: Jul 2011 From: Texas Posts: 2,921 Thanks: 1518 
Tabular integration ... $\displaystyle \int x^5 \cos(3x) \, dx = \color{red}{x^5 \cdot \dfrac{\sin(3x)}{3}} \color{blue}{+ 5x^4 \cdot \dfrac{\cos(3x)}{9}} \color{red}{ 20x^3 \cdot \dfrac{\sin(3x)}{27}} \color{blue}{  60x^2 \cdot \dfrac{\cos(3x)}{81}} \color{red}{+ 120x \cdot \dfrac{\sin(3x)}{243}} \color{blue}{+ 120 \cdot \dfrac{\cos(3x)}{729}} + C$ coefficients may be reduced and sine/cosine terms collected if desired ... 
January 19th, 2018, 10:30 AM  #7 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,649 Thanks: 2630 Math Focus: Mainly analysis and algebra  
January 19th, 2018, 11:38 AM  #8 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,934 Thanks: 1128 Math Focus: Elementary mathematics and beyond 
$$\frac12\int x^5(e^{iax}+e^{iax})\,dx$$ $$\frac12\int x^5e^{iax}+x^5e^{iax}\,dx$$ $$u=e^{iax},\quad\frac{du}{ia}=e^{iax}\,dx$$ O.k., I see my error. Repeated IBP is required to integrate the resulting logarithmic expression. 

Tags 
integrals, type, xncosax, xnsinax 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Type I and Type II Errors  arlobear  Advanced Statistics  2  September 29th, 2016 05:19 PM 
what type of problem is this?  xamdarb  Calculus  1  March 10th, 2014 05:36 PM 
how to type  ranna  New Users  1  February 26th, 2011 03:18 AM 
Dot Product type  wonderboy1953  Number Theory  1  December 31st, 1969 04:00 PM 