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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: 850 Thanks: 327 
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: 7,958 Thanks: 1146 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: 635 Thanks: 401 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: 3,002 Thanks: 1588 
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: 7,675 Thanks: 2655 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: 7,958 Thanks: 1146 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. 

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