Evaluating and Understanding the Integral of ( int_{frac{1}{a}}^{a} frac{x^{n-1}}{1 - x^{4n}} , dx )

int 1/aa frac{x^{n-1}}{1 - x^{4n}} , dx is a definite integral that requires careful evaluation. This integral appears in various applications of mathematical analysis, particularly in areas such as calculus and differential equations. In this article, we will explore a step-by-step method to evaluate this integral using transformation and substitution techniques.

Transformation of the Integral

Consider the integral:

[ I int_{1/a}^a frac{x^{n-1}}{1 - x^{4n}} , dx ]

Let's make the substitution ( x frac{1}{u} ), which implies ( dx -frac{1}{u^2} , du ). This leads to:

[ I int_a^{1/a} frac{left(frac{1}{u}right)^{n-1}}{1 - left(frac{1}{u}right)^{4n}} cdot frac{-1}{u^2} , du int_{1/a}^a frac{u^{-n-1}}{1 - u^{-4n}} cdot (-u^{-2}) , du int_{1/a}^a frac{u^{3n-1}}{u^{4n} - 1} , du ]

Thus, we have two expressions for ( I ):

[ 2I int_{1/a}^a frac{x^{n-1} - x^{3n-1}}{1 - x^{4n}} , dx ]

Let's denote this integral by ( J ).

Substitution and Completing the Square

Next, we make the substitution ( t x^n ), which implies ( dt nx^{n-1} , dx ) and ( x t^{1/n} ). The limits of integration transform as follows: when ( x frac{1}{a} ), ( t left(frac{1}{a}right)^n a^{-n} ); when ( x a ), ( t a^n ).

Revising the integral, we have:

[ J int_{1/a}^a frac{x^{n-1} - x^{3n-1}}{1 - x^{4n}} , dx int_{1/a}^a frac{1 - t^2}{1 - t^4} cdot t^{-1} , frac{dt}{n} ]

This simplifies to:

[ J frac{1}{n} int_{1/a^n}^{a^n} frac{1 - t^2}{1 - t^4} cdot t^{1 - 1/n} , dt frac{1}{n} int_{1/a^n}^{a^n} frac{1 - t^{2/n}}{(1 - t^2)(1 t^2)} , dt ]

Further simplification yields:

[ J frac{1}{n} int_{1/a^n}^{a^n} frac{1 - t^{-2/n}}{t - t^{-1}^2 2} , dt frac{1}{2nsqrt{2}} arctanleft( frac{t - t^{-1}}{sqrt{2}} right) Bigg|_{1/a^n}^{a^n} ]

Therefore, the evaluated integral is:

[ I frac{1}{2nsqrt{2}} arctanleft( frac{a^n - a^{-n}}{sqrt{2}} right) ]

Conclusion

Thus, the definite integral ( int_{1/a}^a frac{x^{n-1}}{1 - x^{4n}} , dx ) is given by:

[ boxed{ int_{1/a}^a frac{x^{n-1}}{1 - x^{4n}} , dx frac{1}{2nsqrt{2}} arctanleft( frac{a^n - a^{-n}}{sqrt{2}} right) } ]

Additional Insight

For the more general case, integrating from (0) to (infty), the integral can be expressed as:

[ boxed{ int_0^infty frac{x^{n-1}}{1 - x^{4n}} , dx frac{pi}{2nsqrt{2}} } ]