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$\sqrt{x^2+a^2}$ içeren integraller

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


2.
$\displaystyle \int\displaystyle \frac{x\,dx}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}$


3.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=\displaystyle \frac{x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{2}-\displaystyle \frac{a^{\displaystyle2}}{2}\ln\lef$


4.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{3}-a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}$


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


6.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=-\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{a^{\displaystyle2}x}$


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


8.
$\displaystyle \int\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\,dx=\displaystyle \frac{x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{2}+\displaystyle \frac{a^{\displaystyle2}}{2}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)$


9.
$\displaystyle \int x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{3}$


10.
$\small \displaystyle \int x^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\,dx=\displaystyle \frac{x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{4}\;-\;\displaystyle \frac{a^{\displaystyle2}x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{8}\;-\;\displaystyle \frac{a^{\displaystyle4}}{8}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)$


11.
$\displaystyle \int x^{\displaystyle3}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle5/2}}{5}-\displaystyle \frac{a^{\displaystyle2}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{3}$


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


13.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x^{\displaystyle2}}\,dx= -\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}+\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)$


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


15.
$\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{x}{a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}$


16.
$\displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{-1}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}$


17.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{-x}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}+\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)$


18.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}+\displaystyle \frac{a^{\displaystyle2}}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}$


19.
$\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{1}{a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}-\displaystyle \frac{1}{a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$


20.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=-\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{a^{\displaystyle4}x}\;-\;\displaystyle \frac{x}{a^{\displaystyle4}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}$


21.
$\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{-1}{2a^{\displaystyle2}x^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}\;-\;\displaystyle \frac{3}{2a^{\displaystyle4}\displaystyle \sqrt{x^{\displaystyle2}\;+\;a^{\displaystyle2}}}+\displaystyle \frac{3}{2a^{\displaystyle5}}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$


22.
$\small \displaystyle \int(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}\,dx=\displaystyle \frac{x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{4}\;+\;\displaystyle \frac{3a^{\displaystyle2}x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{8}\;+\;\displaystyle \frac{3}{8}a^{\displaystyle4}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)$


23.
$\displaystyle \int x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle5/2}}{5}$


24.
$\begin{array}{lcl} \displaystyle \int x^{\displaystyle2}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}\,dx&=&\displaystyle \frac{x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle5/2}}{6}\; -\;\displaystyle \frac{a^{\displaystyle2}x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{24}\;\\ \\ && -\;\displaystyle \frac{a^{\displaystyle4}x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{16}\;-\;\displaystyle \frac{a^{\displaystyle6}}{16}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right) \end{array}$


25.
$\displaystyle \int x^{\displaystyle3}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle7/2}}{7}\;-\;\displaystyle \frac{a^{\displaystyle2}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle5/2}}{5}$


26.
$\small \displaystyle \int\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{x}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{3}\;+\;a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\;-\;a^{\displaystyle3}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$


27.
$\small \displaystyle \int\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{x^{\displaystyle2}}\,dx=-\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{x}\;+\;\displaystyle \frac{3x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{2}\;+\;\displaystyle \frac{3}{2}a^{\displaystyle2}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)$


28.
$\displaystyle \int\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{x^{\displaystyle3}}\,dx=-\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{2x^{\displaystyle2}}\;+\;\displaystyle \frac{3}{2}\d$


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