<s>[INST] How are these steps in this integral done? $$\int\frac{du}{u^{2}-2}=\frac{1}{2\sqrt{2}}\int\frac{du}{u-\sqrt{2}}-\int\frac{du}{u+\sqrt{2}}=\frac{1}{2\sqrt{2}}\ln\left|\frac{u-\sqrt{2}}{u+\sqrt{2}}\right|$$
How to do the middle step?
 [/INST] $$\int\frac{du}{u^{2}-2}=\frac{1}{2\sqrt{2}} \left(\int\frac{du}{u-\sqrt{2}}-\int\frac{du}{u+\sqrt{2}}\right )=\frac{1}{2\sqrt{2}}\ln\left|\frac{u-\sqrt{2}}{u+\sqrt{2}}\right|$$
Because of this...
$$\frac 1 {a^2-b^2}=\frac 1 {2b} \left (\frac 1 {a-b}-\frac 1 {a+b} \right)$$
$$\int \frac {da} {a^2-b^2}=\frac 1 {2b} \left (\int \frac 1 {a-b}da-\int \frac 1 {a+b}da \right)$$
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