List of Reciprocal integral rules

In integral calculus, there are some notable integral rules in which the mathematical functions are in reciprocal form (multiplicatively inverse) and they are used as formulas in integration. So, let’s learn the list of reciprocal integration rules with proofs before using them in definite and indefinite integration problems.

Basic Reciprocal integral rule

$\displaystyle \int{\dfrac{1}{x}\,}dx$ $\,=\,$ $\log_{e}{x}+c$

The integral of the reciprocal of a variable is equal to the natural logarithm of variable plus the integral constant.

Reciprocal integral rule in One plus square

$\displaystyle \int{\dfrac{1}{1+x^2}\,}dx$ $\,=\,$ $\arctan{x}+c$ (or) $\tan^{-1}{x}+c$

Reciprocal integral rule in Difference of squares

$\displaystyle \int{\dfrac{1}{x^2-a^2}\,}dx$ $\,=\,$ $\dfrac{1}{2a}\log_{e}{\Bigg|\dfrac{x-a}{x+a}\Bigg|}+c$

Reciprocal integral rule in Sum of squares

$\displaystyle \int{\dfrac{1}{x^2+a^2}\,}dx$ $\,=\,$ $\dfrac{1}{a}\arctan{\Big(\dfrac{x}{a}\Big)}+c$ (or) $\dfrac{1}{a}\tan^{-1}{\Big(\dfrac{x}{a}\Big)}+c$

Reciprocal integral rule in Square root of one minus square

$\displaystyle \int{\dfrac{1}{\sqrt{1-x^2}}\,}dx$ $\,=\,$ $\arcsin{x}+c$ (or) $\sin^{-1}{x}+c$

The integral of one by square root of one minus square of a variable is equal to the inverse sine of the variable plus the integral constant.

Reciprocal Integral rule in Product of variable and Square root of square minus one

$\displaystyle \int{\dfrac{1}{|x|\sqrt{x^2-1}}\,}dx$ $\,=\,$ $\operatorname{arcsec}{x}+c$ (or) $\sec^{-1}{x}+c$