Can anyone provide a hint for solving this definite integral: $\int_{-a}^{a}\frac{xdy}{(x^{2}+y^{2})^{\frac{3}{2}}}$
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See also: Calculating $\int_{-a}^{a} \frac{x\cdot dy}{(x^2+y^2)^{3/2}}$ requires unusual substitution? – Martin Sleziak Apr 13 '19 at 22:37
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hint: you may start from $y=x \tan(\theta)$
Math-fun
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4That is the starting step. Try that change of variable you will see that the integral becomes simple. – Rogelio Molina Jun 09 '15 at 06:21
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Sorry, I shouldn't be that lazy :-D. Got the answer btw. Thanks a lot :-) – bpr3003 Jun 09 '15 at 06:29
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Another way, in this particular example, is to rewrite your expression as $$ \begin{aligned} \frac{x}{(x^2+y^2)^{3/2}}&=\frac{x^2+y^2-y^2}{x(x^2+y^2)^{3/2}}=\frac{1}{x(x^2+y^2)^{1/2}}-\frac{y^2}{x(x^2+y^2)^{3/2}}\\ &=\Bigl(\frac{d}{dy} y\Bigr)\cdot \frac{1}{x(x^2+y^2)^{1/2}}+ y\cdot \frac{d}{dy}\frac{1}{x(x^2+y^2)^{1/2}}\\ &=\frac{d}{dy}\Bigl(\frac{y}{x(x^2+y^2)^{1/2}}\Bigr) \end{aligned} $$ Thus, you have a simple primitive function that you can put into the fundamental theorem of calculus.
mickep
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