The irrational functions are come in integral calculus and it is not possible to find the integration of the irrational functions with the standard integral rules. Hence, it requires some special integral formulas in some cases to evaluate the integrals of the irrational functions. The following are the integral rules of the irrational functions with proofs.

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

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

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

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