Integrating a Tricky Rational Function with Substitution

[utkarshakash]

Homework Statement

$\displaystyle \int^∞_0 \dfrac{dx}{a^2 + \left(x-\frac{1}{x} \right)^2}$ a>=2

The Attempt at a Solution

$\displaystyle \int^∞_0 \dfrac{x^2 dx}{x^2a^2 + (x^2-1)^2}$

 

[Saitama]

Homework Statement

$\displaystyle \int^∞_0 \dfrac{dx}{a^2 + \left(x-\frac{1}{x} \right)^2}$ a>=2

The Attempt at a Solution

$\displaystyle \int^∞_0 \dfrac{x^2 dx}{x^2a^2 + (x^2-1)^2}$

Put $\displaystyle x=\frac{1}{t}$ in the given integral.

 

[sankalpmittal]

Homework Statement

$\displaystyle \int^∞_0 \dfrac{dx}{a^2 + \left(x-\frac{1}{x} \right)^2}$ a>=2

The Attempt at a Solution

$\displaystyle \int^∞_0 \dfrac{x^2 dx}{x^2a^2 + (x^2-1)^2}$

Not sure about Pranav's hint but here is how I would have done it:

Making a tricky substitution of,

1/t = x-1/x

OR

In your attempt at solution, expand the denominator, then write numerator as x²-1+1, break the denominator, then in each integrand, divide both sides by x², try making the denominator the perfect square, then in term like Y² in the denominator, let Y=t...etc..