Some Special Integrals

(i) \[\int\frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C\]

(ii) \[\int\frac{dx}{a^{2}-x^{2}}=\frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+C\]

(iii) \[\int\frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}+C\]

(iv) \[\int\frac{dx}{\sqrt{x^{2}-a^{2}}}=\log|x+\sqrt{x^{2}-a^{2}}|+C\]

(v) \[\int\frac{dx}{\sqrt{a^{2}-x^{2}}}=\sin^{-1}\frac{x}{a}+C\]

(vi) \[\int\frac{dx}{\sqrt{x^{2}+a^{2}}}=\log|x+\sqrt{x^{2}+a^{2}}|+C\]