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I am currently studying a course on Perturbation Methods and in particular an example considering the following integral [tex] \int_{0}^{\frac{\pi}{4}} \frac{d\theta}{\epsilon^2 + \sin^2 \theta}. [/tex]

There's a section of the working where, having used the Taylor expansion of sin near 0 and using sin θ ≈ θ together with substitution θ=εu, we get the following fraction for the integrand

[tex] \frac{1}{\epsilon^2 + \epsilon^2 u^2 - \frac13 \epsilon^4 u^4 + \cdots}.[/tex]

This then in both my lecture notes and a book I'm following becomes

[tex] \frac{1}{\epsilon^2}\left( \frac{1}{1+u^2} + \frac{\epsilon u^4}{3(1+u^2)^2} + \cdots \right). [/tex]

Can anyone see how these are equal?

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# From a fraction with infinite sum in denominator to partial fractions?

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