How can I find the integral of this expression: $$\int_{-a}^{a}\frac{x*dy}{( x^2 + y^2 )^{\frac{3}{2}}}$$
Please would yo people give me a step by step solution. I am stuck in this
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i think there is no elementary solution – Dr. Sonnhard Graubner Feb 17 '17 at 19:44
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Since the integral is with respect to $dy$, do you mean that $x$ is simply a constant (and thus can be factored out)? Or is there something else, some other relation, that you didn't mention? – zipirovich Feb 17 '17 at 19:47
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- factor out the $x's$ and substitute $u=y/x$ so that you see that you just need to find $\int du/(1+u^2)^{3/4}.$ 2. Plug that into mathematica and see that the answer is in terms of hypergeometric functions. 3. Ask yourself 'have I written the problem down correctly?'
– spaceisdarkgreen Feb 17 '17 at 19:52 -
1I am sorry people I made an error in the definition of the integral. Now I have just made an edit. Yes @zipirovich x is just a constant. – billyandriam Feb 17 '17 at 20:05
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1thank you @spaceisddarkgreen ! Yes there was a mistake in the problem I wrote! sorry – billyandriam Feb 17 '17 at 20:05
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Some other posts about the same integral - found using Approach0: Calculating $\int_{-a}^{a} \frac{x\cdot dy}{(x^2+y^2)^{3/2}}$ requires unusual substitution? and Hint for solving a definite integral $\int_{-a}^{a}\frac{xdy}{(x^{2}+y^{2})^{\frac{3}{2}}}$. Some tips on searching: How to search on this site? – Martin Sleziak Apr 13 '19 at 22:36
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@MartinSleziak Thank you for taking the time to look at my posts and keep math.se a nice spam-free place. Can you give me a suggestion what to do with this post? It has duplicates and I cannot delete it since it has answers. Do you have the privileges on your end to delete it? I cannot reword it nor add context since it came right as such, as a question in my calculus book. – billyandriam Apr 13 '19 at 23:48
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@billyandr Including the source of the problem is a part of a way to provide context, see Provide Context and Include source / motivation for your question. So perhaps it is worth adding. I am not really sure it is necessary to delete this post - of course, other might have a different opinion. – Martin Sleziak Apr 13 '19 at 23:52
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@MartinSleziak Ok I will add the source. But what I am saying is from my end, I just cannot delete it. I get an error message saying that I cannot delete it since it has answers and other people have invested time in answering. So this one issue is entirely up to the moderators. What's left for me now is be wiser next time before posting a question, use the tools you provided to search for my duplicates and therefore avoid reasking a solved issue. – billyandriam Apr 14 '19 at 00:01
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As suggested, we have
$$\begin{align} \int_{y=-a}^{a} \frac{x\,dy}{(x^{2}+y^{2})^{\frac{3}{2}}} &\,=\, \frac{1}{x^{2}}\int_{y=-a}^{a} \frac{dy}{(1+\left(\frac{y}{x}\right)^{2})^{\frac{3}{2}}} \,=\, \frac{1}{x}\int_{u=-\frac{a}{x}}^{\frac{a}{x}} \frac{du}{(1+u^{2})^{\frac{3}{2}}} \\[0.3cm] &\,=\, \left.\frac{1}{x}\cdot \frac{u}{\sqrt{u^{2}+1}}\right|_{u=-\frac{a}{x}}^{\frac{a}{x}} \end{align}$$
using the substitution $y=ux$. You should be able to take it from there.
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