Integration solving problem

Question

A integration is given $$x-x_0 = \pm \int_{0}^{\phi(x)}\frac{d\Phi}{\sqrt\frac{\lambda}{2}(\Phi^2-\frac{m^2}{\lambda})} \tag{1}$$ The author said that, equation (2) can be written from equation (1) by inverting; $$\Phi(x)= \pm \left(\frac{m}{\sqrt\lambda }\right)\tanh\left[\left(\frac{m}{\sqrt2}\right)(x-x_0)\right] \tag{2} $$

How to solve this integration easily?

Anyone can help?

Answer 1

Hint: Use

$$\int \frac{1}{a^2 -b^2} da = - \frac{\tanh^{-1}(a/b)}{b} + \text{constant}.$$

Answer 2

Partial fractions: $$ \frac{d\Phi}{\Phi^2-\frac{m^2}{\lambda}} = \left(\frac{A}{\Phi-\frac{m}{\sqrt{\lambda}}}+\frac{B}{\Phi+\frac{m}{\sqrt{\lambda}}}\right)\,d\Phi. $$ Use some algebra to figure out what numbers $A$ and $B$ are. When you integrate, you get $$ A\log_e\left(\Phi-\frac{m}{\sqrt{\lambda}}\right) + B\log_e\left(\Phi+\frac{m}{\sqrt{\lambda}}\right). $$ Then use identities involving logarithms to write it as just one logarithm.