Integrate the indefinite integral?

Question

Integrate the indefinite integral ?

$$\int \frac{x}{\sqrt{x^2-a^2} + \sqrt{x^2+a^2}}dx$$

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My try :-

Let $x^2 = t \to 2xdx=dt$ and then rationalize

I have,

$$\frac{1}{4a^2}\int \sqrt{t-a^2}- \sqrt{t+a^2}dt$$

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I know its silly, but I don't know how to get further ?

Answer 1

For example, $$ \int(t-a^2)^{1/2}\,dt=\frac{2}{3}(t-a^2)^{3/2}+C $$

Answer 2

Rationalizing yields $\int\frac{x\left(\sqrt{x^2+a^2}-\sqrt{x^2-a^2}\right)}{2a^2}$ using your substitution the integral becomes $\color{red}{\frac{1}{4a^2}}\displaystyle\int\sqrt{t+a^2}-\sqrt{t-a^2}\,\mathrm{d}x$. Because the derivative of $t+$ any constant is just $1$, we simply integrate, $\frac{1}{4a^2}\left(\frac{2}{3}(t+a^2)^{3/2}-\frac{2}{3}(t-a^2)^{3/2}\right)+C$ and then substitute back $t=x^2$