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ΟΛΟΚΛΗΡΩΜΑΤΑ ΠΟΥ ΠΕΡΙΕΧΟΥΝ ΤΟ a² - x²

Υποθέτουμε ότι x²<a²:

1.: $\displaystyle \int\displaystyle \frac{dx}{a^{\displaystyle2}-x^{\displaystyle2}}=\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{a+x}{a-x}\right)\;\;$ or $\;\;\displaystyle \frac{1}{a}\tanh^{\displaystyle-1}\displaystyle \frac{x}{a}$
2.: $\displaystyle \int\displaystyle \frac{x\,dx}{a^{\displaystyle2}-x^{\displaystyle2}}=-\displaystyle \frac{1}{2}\ln(a^{\displaystyle2}-x^{\displaystyle2})$
3.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{a^{\displaystyle2... ...le2}}=-x+\displaystyle \frac{a}{2}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$
4.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{a^{\displaystyle2... ...laystyle \frac{a^{\displaystyle2}}{2}\ln(a^{\displaystyle2}-x^{\displaystyle2})$
5.: $\displaystyle \int\displaystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle... ...aystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)$
6.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(a^{\displaystyle2}... ...tyle \frac{1}{2a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$
7.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(a^{\displaystyle2}... ...aystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)$
8.: $\displaystyle \int\displaystyle \frac{dx}{(a^{\displaystyle2}-x^{\displaystyle2... ...tyle \frac{1}{4a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$
9.: $\displaystyle \int\displaystyle \frac{x\,dx}{(a^{\displaystyle2}-x^{\displaysty... ...isplaystyle2}}=\displaystyle \frac{1}{2(a^{\displaystyle2}-x^{\displaystyle2})}$
10.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(a^{\displaystyle... ...yle2})}-\displaystyle \frac{1}{4a}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$
11.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(a^{\displaystyle... ...aystyle2})}+\displaystyle \frac{1}{2}\ln(a^{\displaystyle2}-x^{\displaystyle2})$
12.: $\displaystyle \int\displaystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle... ...aystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)$
13.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(a^{\displaystyle2}... ...tyle \frac{3}{4a^{\displaystyle5}}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$
14.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(a^{\displaystyle2}... ...aystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)$
15.: $\displaystyle \int\displaystyle \frac{dx}{(a^{\displaystyle2}-x^{\displaystyle2... ...laystyle \frac{dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}$
16.: $\displaystyle \int\displaystyle \frac{x\,dx}{(a^{\displaystyle2}-x^{\displaysty... ...yle \frac{1}{2(n-1)(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}$
17.: $\displaystyle \int\displaystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle... ...aystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}$
18.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle m}\,dx}{(a^{\displaystyl... ...laystyle m-2}\,dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}$
19.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(a^{\displaystyle2... ...x^{\displaystyle m-2}(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n}}$