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chomsky
 Post subject: Simple integration problems
PostPosted: Wed, 23 May 2012 13:05:07 UTC 
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How do I integrate \frac{x^3}{x^2 - 4} ? . I already know how to integrate basic indefinite integrals, but I don't know how to solve this.
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outermeasure
 Post subject: Re: Simple integration problems
PostPosted: Wed, 23 May 2012 13:38:53 UTC 
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chomsky wrote:
How do I integrate \frac{x^3}{x^2 - 4} ? . I already know how to integrate basic indefinite integrals, but I don't know how to solve this.


First perform long division.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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chomsky
 Post subject: Re: Simple integration problems
PostPosted: Wed, 23 May 2012 14:13:59 UTC 
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Do you mean that I should divide x^3 by x^2 - 4. I tried that, but I'm getting a remainder. The whole thing comes out to x + \frac{4x}{x^2-4}. The remainder isn't of the form \frac{dx}{x^2 \pm a^2}, so I don't know how to proceed further.
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outermeasure
 Post subject: Re: Simple integration problems
PostPosted: Wed, 23 May 2012 14:25:05 UTC 
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chomsky wrote:
Do you mean that I should divide x^3 by x^2 - 4. I tried that, but I'm getting a remainder. The whole thing comes out to x + \frac{4x}{x^2-4}. The remainder isn't of the form \frac{dx}{x^2 \pm a^2}, so I don't know how to proceed further.


Hint: What is \displaystyle\int\frac{f'(x)}{f(x)}\,\mathrm{d}x?

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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chomsky
 Post subject: Re: Simple integration problems
PostPosted: Wed, 23 May 2012 16:42:36 UTC 
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I don't know.
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Shadow
 Post subject: Re: Simple integration problems
PostPosted: Wed, 23 May 2012 19:00:21 UTC 
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chomsky wrote:
I don't know.


Then make a u-substitution.

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