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$\sqrt{a^2-x^2}$ içeren integraller

1.
$\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}=\sin^{\displaystyle-1}\displaystyle \frac{x}{a}$


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


3.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}=-\displaystyle \frac{x\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{2}\;+\;\displaystyle \frac{a^{\displaystyle2}}{2}\sin^{\displaystyle-1}\displaystyle \frac{x}{a}$


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


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


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


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


8.
$\displaystyle \int\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}\,dx=\displaystyle \frac{x\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{2}\;+\;\displaystyle \frac{a^{\displaystyle2}}{2}\sin^{\displaystyle-1}\displaystyle \frac{x}{a}$


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


10.
$\begin{array}{lcl} \displaystyle \int x^{\displaystyle2}\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}\,dx&=&-\displaystyle \frac{x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle3/2}}{4}\;+\;\displaystyle \frac{a^{\displaystyle2}x\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{8}\;\\\\&&+\;\displaystyle \frac{a^{\displaystyle4}}{8}\sin^{\displaystyle-1}\displaystyle \frac{x}{a} \end{array}$


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


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


13.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x^{\displaystyle2}}\,dx=-\displaystyle \frac{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x}-\sin^{\displaystyle-1}\displaystyle \frac{x}{a}$


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


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


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


17.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{x}{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}-\sin^{\displaystyle-1}\displaystyle \frac{x}{a}$


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


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


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


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


22.
$\begin{array}{lcl} \displaystyle \int(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}\,dx&=&\displaystyle \frac{x(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}{4}\;+\;\displaystyle \frac{3a^{\displaystyle2}x\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{8}\;\\\\&&+\;\displaystyle \frac{3}{8}a^{\displaystyle4}\sin^{\displaystyle-1}\displaystyle \frac{x}{a} \end{array}$


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


24.
$\begin{array}{ll} \displaystyle \int x^{\displaystyle2}(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}\,dx= -\displaystyle \frac{x(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle5/2}}{6}\;+\;\displaystyle \frac{a^{\displaystyle2}x(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}{24}\\+\displaystyle \frac{a^{\displaystyle4}x\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{16}\;+\;\displaystyle \frac{a^{\displaystyle6}}{16}\sin^{\displaystyle-1}\displaystyle \frac{x}{a} \end{array}$


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


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


27.
$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}{x^{\displaystyle2}}\,dx&=&-\displaystyle \frac{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}{x}\;-\;\displaystyle \frac{3x\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{2}\;\\\\&&-\;\displaystyle \frac{3}{2}a^{\displaystyle2}\sin^{\displaystyle-1}\displaystyle \frac{x}{a} \end{array}$


28.
$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}{x^{\displaystyle3}}\,dx&=& -\displaystyle \frac{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}{2x^{\displaystyle2}}\;-\;\displaystyle \frac{3\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{2}\;\\\\&&+\;\displaystyle \frac{3}{2}a\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x}\right) \end{array}$


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