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$x^3+a^3$ içeren integraller

1.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}+a^{\displaystyle3}}=\displaystyle \frac{1}{6a^{\displaystyle2}}\ln\displaystyle \frac{(x+a)^{\displaystyle2}}{x^{\displaystyle2}-ax+a^{\displaystyle2}}\,+\,\displaystyle \frac{1}{a^{\displaystyle2}\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}}$


2.
$\displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}=\displaystyle \frac{1}{6a}\ln\displaystyle \frac{x^{\displaystyle2}-ax+a^{\displaystyle2}}{(x+a)^{\displaystyle2}}\,+\,\displaystyle \frac{1}{a\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}}$


3.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}=\displaystyle \frac{1}{3}\ln(x^{\displaystyle3}+a^{\displaystyle3})$


4.
$\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle3}+a^{\displaystyle3})}=\displaystyle \frac{1}{3a^{\displaystyle3}}\ln\left(\displaystyle \frac{x^{\displaystyle3}}{x^{\displaystyle3}+a^{\displaystyle3}}\right)$


5.
$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle3}+a^{\displaystyle3})}&=&-\displaystyle \frac{1}{a^{\displaystyle3}x}\,-\,\displaystyle \frac{1}{6a^{\displaystyle4}}\ln\displaystyle \frac{x^{\displaystyle2}-ax+a^{\displaystyle2}}{(x+a)^{\displaystyle2}}\,\\\\&&-\,\displaystyle \frac{1}{a^{\displaystyle4}\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}} \end{array}$


6.
$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}&=&\displaystyle \frac{x}{3a^{\displaystyle3}(x^{\displaystyle3}+a^{\displaystyle3})}\,+\,\displaystyle \frac{1}{9a^{\displaystyle5}}\ln\displaystyle \frac{(x+a)^{\displaystyle2}}{x^{\displaystyle2}-ax+a^{\displaystyle2}}\,\\\\&&+\,\displaystyle \frac{2}{3a^{\displaystyle5}\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}} \end{array}$


7.
$\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}&=&\displaystyle \frac{x^{\displaystyle2}}{3a^{\displaystyle3}(x^{\displaystyle3}+a^{\displaystyle3})}\,+\,\displaystyle \frac{1}{18a^{\displaystyle4}}\ln\displaystyle \frac{x^{\displaystyle2}-ax+a^{\displaystyle2}}{(x+a)^{\displaystyle2}}\,\\\\&&+\,\displaystyle \frac{1}{3a^{\displaystyle4}\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}} \end{array}$


8.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}=-\displaystyle \frac{1}{3(x^{\displaystyle3}+a^{\displaystyle3})}$


9.
$\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}=\displaystyle \frac{1}{3a^{\displaystyle3}(x^{\displaystyle3}+a^{\displaystyle3})}\,+\,\displaystyle \frac{1}{3a^{\displaystyle6}}\ln\left(\displaystyle \frac{x^{\displaystyle3}}{x^{\displaystyle3}+a^{\displaystyle3}}\right)$


10.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}=-\displaystyle \frac{1}{a^{\displaystyle6}x}\,-\,\displaystyle \frac{x^{\displaystyle2}}{3a^{\displaystyle6}(x^{\displaystyle3}+a^{\displaystyle3})}\,-\,\displaystyle \frac{4}{3a^{\displaystyle6}}\displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}$


11.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle m}\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}=\displaystyle \frac{x^{\displaystyle m-2}}{m-2}\,-\,a^{\displaystyle3}\displaystyle \int\displaystyle \frac{x^{\displaystyle m-3}\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}$


12.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle n}(x^{\displaystyle3}+a^{\displaystyle3})}=\displaystyle \frac{-1}{a^{\displaystyle3}(n-1)x^{\displaystyle n-1}}-\displaystyle \frac{1}{a^{\displaystyle3}}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle n-3}(x^{\displaystyle3}+a^{\displaystyle3})}$


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