# Integral Rules of Algebraic functions

### Power rule

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

### Reciprocal rules

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

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

$(3)\,\,\,$ $\displaystyle \int{\dfrac{1}{1+x^2}\,}dx$ $\,=\,$ $\tan^{-1}{x}+c$

$(4)\,\,\,$ $\displaystyle \int{\dfrac{1}{|x|\sqrt{x^2-1}}\,}dx$ $\,=\,$ $\sec^{-1}{x}+c$

$(5)\,\,\,$ $\displaystyle \int{\dfrac{1}{x^2-a^2}\,}dx$ $\,=\,$ $\dfrac{1}{2a}\log_{e}{\Bigg|\dfrac{x-a}{x+a}\Bigg|}+c$

$(6)\,\,\,$ $\displaystyle \int{\dfrac{1}{x^2+a^2}\,}dx$ $\,=\,$ $\dfrac{1}{a}\tan^{-1}{\Big(\dfrac{x}{a}\Big)}+c$

### Exponential rules

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

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

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