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$x^2-a^2$ içeren integraller

$x^2>a^2\quad \text{kabul edecektir}$


1.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\;\; \text{veya}\;\;-\displaystyle \frac{1}{a}coth^ {\displaystyle-1}\displaystyle \frac{x}{a}$


2.
$\displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{1}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})$


3.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=x+\displaystyle \frac{a}{2}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$


4.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{x^{\displaystyle2}}{2}+\displaystyle \frac{a^{\displaystyle2}}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})$


5.
$\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{2a^{\displaystyle2}}\ln\left(\displaystyle \frac{x^{\displaystyle2}-a^{\displaystyle2}}{x^{\displaystyle2}}\right)$


6.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{a^{\displaystyle2}x}+\displaystyle \frac{1}{2a^{\displaystyle3}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$


7.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{2a^{\displaystyle2}x^{\displaystyle2}}-\displaystyle \frac{1}{2a^{\displaystyle4}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)$


8.
$\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-x}{2a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}-\displaystyle \frac{1}{4a^{\displaystyle3}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$


9.
$\displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-1}{2(x^{\displaystyle2}-a^{\displaystyle2})}$


10.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-x}{2(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{4a}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$


11.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-a^{\displaystyle2}}{2(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})$


12.
$\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-1}{2a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{2a^{\displaystyle4}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)$


13.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=-\displaystyle \frac{1}{a^{\displaystyle4}x}-\displaystyle \frac{x}{2a^{\displaystyle4}(x^{\displaystyle2}-a^{\displaystyle2})}-\displaystyle \frac{3}{4a^{\displaystyle5}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$


14.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=-\displaystyle \frac{1}{2a^{\displaystyle4}x^{\displaystyle2}}-\displaystyle \frac{1}{2a^{\displaystyle4}(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{a^{\displaystyle6}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)$


15.
$\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-x}{2(n-1)a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}-\displaystyle \frac{2n-3}{(2n-2)a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}$


16.
$\displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-1}{2(n-1)(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}$


17.
$\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-1}{2(n-1)a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}-\displaystyle \frac{1}{a^{\displaystyle2}}\d$


18.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle m}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \int\displaystyle \frac{x^{\displaystyle m-2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}+a^{\displaystyle2}\displaystyle \int\displaystyle \frac{x^{\displaystyle m-2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}$


19.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{1}{a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m-2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}-\displaystyle \frac{1}{a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}$


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