Math Doubts

Integral Rules of Algebraic functions

There are three types of algebraic integral rules in integral calculus. They are used as formulas in calculating both indefinite and definite integrals of the algebraic functions. So, let’s learn each algebraic integration formula with proof to know how to use them in indefinite and definite integration problems.

Power rule

$\displaystyle \int{x^n\,}dx$ $\,=\,$ $\dfrac{x^{n+1}}{n+1}+c$

Exponential integral rules

The following two exponential integral rules are the integration formulae in which the algebraic functions are in exponential form.

$(1).\,\,\,$ $\displaystyle \int{a^x\,}dx$ $\,=\,$ $\dfrac{a^x}{\log_{e}{a}}+c$

$(2).\,\,\,$ $\displaystyle \int{e^x\,}dx$ $\,=\,$ $e^x+c$

Reciprocal integral rules

The following six reciprocal integral rules are the integration formulas in which the algebraic functions are in multiplicative inverse form.

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

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

$(3).\,\,\,$ $\displaystyle \int{\dfrac{1}{x^2-a^2}\,}dx$ $\,=\,$ $\dfrac{1}{2a}\log_{e}{\Bigg|\dfrac{x-a}{x+a}\Bigg|}+c$

$(4).\,\,\,$ $\displaystyle \int{\dfrac{1}{x^2+a^2}\,}dx$ $\,=\,$ $\dfrac{1}{a}\arctan{\Big(\dfrac{x}{a}\Big)}+c$ (or) $\dfrac{1}{a}\tan^{-1}{\Big(\dfrac{x}{a}\Big)}+c$

$(5).\,\,\,$ $\displaystyle \int{\dfrac{1}{\sqrt{1-x^2}}\,}dx$ $\,=\,$ $\arcsin{x}+c$ (or) $\sin^{-1}{x}+c$

$(6).\,\,\,$ $\displaystyle \int{\dfrac{1}{|x|\sqrt{x^2-1}}\,}dx$ $\,=\,$ $\operatorname{arcsec}{x}+c$ (or) $\sec^{-1}{x}+c$

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