I-0-x-a-2-b-2-x-2a-2-x-2-2b-2-dx-a-2-x-2-b-2-2-a-2-b-2-x-a-2-x-2-b-2-

Question Number 141244 by ajfour last updated on 17/May/21

I = \int_{0}^{\infty} \frac{x {(a^{2} - b^{2}) x - 2 a^{2} x^{2} - 2 b^{2}} d x}{{(a^{2} x^{2} + b^{2})}^{2} {(a^{2} - b^{2}) x + a^{2} x^{2} + b^{2}}}

Commented by ajfour last updated on 17/May/21

this will get me perimeter of ellipse!

Answered by MJS_new last updated on 17/May/21

I_{1} = \int \frac{x}{{(a^{2} x^{2} + b^{2})}^{2}} d x = - \frac{1}{2 a^{2} (a^{2} x^{2} + b^{2})}

I_{2} = - \frac{3}{a^{2} - b^{2}} \int \frac{d x}{a^{2} x^{2} + b^{2}} = - \frac{3 \arctan \frac{a x}{b}}{a (a^{2} - b^{2}) b}

I_{3} = \frac{3}{a^{2} - b^{2}} \int \frac{d x}{a^{2} x^{2} + (a^{2} - b^{2}) x + b^{2}} =

= \frac{3}{(a^{2} - b^{2}) \sqrt{a^{4} - 6 a^{2} b^{2} + b^{4}}} \ln \frac{2 a^{2} x^{2} + a^{2} - b^{2} - \sqrt{a^{4} - 6 a^{2} b^{2} + b^{4}}}{2 a^{2} x^{2} + a^{2} - b^{2} + \sqrt{a^{4} - 6 a^{2} b^{2} + b^{4}}}

now calculate within the borders

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