Integrals of Some Particular Functions

1) ${\displaystyle \int \frac{dx}{x^2-a^2}=\frac{1}{2a}\log \left|\frac{x-a}{x+a}\right|+C}$

2) ${\displaystyle \int \frac{dx}{a^2-x^2}=\frac{1}{2a}\log \left|\frac{a+x}{a-x}\right|+C}$

3) ${\displaystyle \int \frac{dx}{x^2-a^2}=\frac{1}{a} {\tan }^{-1}\left(\frac{x}{a}\right)+C}$

4) ${\displaystyle \int \frac{dx}{\sqrt{x^2-a^2}}=\log |x+\sqrt{x^2-a^2}|+C}$

5) ${\displaystyle \int \frac{dx}{\sqrt{a^2-x^2}}={\sin }^{-1}\left(\frac{x}{a}\right)+C}$

6)  ${\displaystyle \int \frac{dx}{\sqrt{x^2+a^2}}=\log |x+\sqrt{x^2+a^2}|+C}$

7) To find the integral ${\displaystyle \int \frac{dx}{a{x}^2+bx+c}}$

8) To find the integral of the type ${\displaystyle \int \frac{dx}{\sqrt{a{x}^2+bx+c}}}$

9) To find the integral of the type ${\displaystyle \int \frac{px+q}{a{x}^2+bx+c}dx}$

10) For the evaluation of the integral of the type ${\displaystyle \int \frac{px+q}{\sqrt{a{x}^2+bx+c}}dx}$