Integrate $\int \frac{dx}{(4x^2-9)^{3/2}}$

Question

How to solve the integral $$\int \frac{dx}{(4x^2-9)^{3/2}}$$

Answer 1

Change the the denominator to $$\left(4x^2-9 \right)^{\frac{3}{2}} = 8 \left( x^2-\frac{9}{4} \right)^{\frac{3}{2}}.$$ Then apply the secant substitution, $$x=\frac{3}{2}\sec \theta,$$ and continue in the usual manner.

Answer 2

Please refer to this.

The answer is $$ -\frac{x}{9\sqrt{4x^2-9}} + c $$ where $c$ is constant.

Answer 3

$x = \frac{3}{2}\cosh t$ should be the obvious substitution to solve it.

$$\int \frac{\frac{3}{2}\sinh t ~dt}{9^{3/2} \sinh^3 dt} = \frac{1}{18} \int\frac{dt}{\sinh^2 t} = \frac{1}{18} \coth t + \mathcal{C} $$

Thus

$$\int\frac{dx}{\left(4x^2 - 9\right)^{3/2}} = \frac{1}{18} \coth\left(\cosh^{-1} \frac{2x}{3}\right) +\mathcal {C}$$