 Matematik
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\(x^3+a^3\) içeren integraller
- 1.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}+a^{\displaystyle3}}=\displaystyle \frac{1}{6a^{\displaystyle2}}\ln\displaystyle \frac{(x+a)^{\displaystyle2}}{x^{\displaystyle2}-ax+a^{\displaystyle2}}\,+\,\displaystyle \frac{1}{a^{\displaystyle2}\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}}\)
- 2.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}=\displaystyle \frac{1}{6a}\ln\displaystyle \frac{x^{\displaystyle2}-ax+a^{\displaystyle2}}{(x+a)^{\displaystyle2}}\,+\,\displaystyle \frac{1}{a\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}}\)
- 3.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}=\displaystyle \frac{1}{3}\ln(x^{\displaystyle3}+a^{\displaystyle3})\)
- 4.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle3}+a^{\displaystyle3})}=\displaystyle \frac{1}{3a^{\displaystyle3}}\ln\left(\displaystyle \frac{x^{\displaystyle3}}{x^{\displaystyle3}+a^{\displaystyle3}}\right)\)
- 5.
- \(\small \begin{array}{lcl}
\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle3}+a^{\displaystyle3})}&=&-\displaystyle \frac{1}{a^{\displaystyle3}x}\,-\,\displaystyle \frac{1}{6a^{\displaystyle4}}\ln\displaystyle \frac{x^{\displaystyle2}-ax+a^{\displaystyle2}}{(x+a)^{\displaystyle2}}\,\\\\&&-\,\displaystyle \frac{1}{a^{\displaystyle4}\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}}
\end{array}\)
- 6.
- \(\small \begin{array}{lcl}
\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}&=&\displaystyle \frac{x}{3a^{\displaystyle3}(x^{\displaystyle3}+a^{\displaystyle3})}\,+\,\displaystyle \frac{1}{9a^{\displaystyle5}}\ln\displaystyle \frac{(x+a)^{\displaystyle2}}{x^{\displaystyle2}-ax+a^{\displaystyle2}}\,\\\\&&+\,\displaystyle \frac{2}{3a^{\displaystyle5}\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}}
\end{array}\)
- 7.
- \(\small \begin{array}{lcl}
\displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}&=&\displaystyle \frac{x^{\displaystyle2}}{3a^{\displaystyle3}(x^{\displaystyle3}+a^{\displaystyle3})}\,+\,\displaystyle \frac{1}{18a^{\displaystyle4}}\ln\displaystyle \frac{x^{\displaystyle2}-ax+a^{\displaystyle2}}{(x+a)^{\displaystyle2}}\,\\\\&&+\,\displaystyle \frac{1}{3a^{\displaystyle4}\displaystyle \sqrt{3}}\tan^{\displaystyle-1}\displaystyle \frac{2x-a}{a\displaystyle \sqrt{3}}
\end{array}\)
- 8.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}=-\displaystyle \frac{1}{3(x^{\displaystyle3}+a^{\displaystyle3})}\)
- 9.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}=\displaystyle \frac{1}{3a^{\displaystyle3}(x^{\displaystyle3}+a^{\displaystyle3})}\,+\,\displaystyle \frac{1}{3a^{\displaystyle6}}\ln\left(\displaystyle \frac{x^{\displaystyle3}}{x^{\displaystyle3}+a^{\displaystyle3}}\right)\)
- 10.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle3}+a^{\displaystyle3})^{\displaystyle2}}=-\displaystyle \frac{1}{a^{\displaystyle6}x}\,-\,\displaystyle \frac{x^{\displaystyle2}}{3a^{\displaystyle6}(x^{\displaystyle3}+a^{\displaystyle3})}\,-\,\displaystyle \frac{4}{3a^{\displaystyle6}}\displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}\)
- 11.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle m}\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}=\displaystyle \frac{x^{\displaystyle m-2}}{m-2}\,-\,a^{\displaystyle3}\displaystyle \int\displaystyle \frac{x^{\displaystyle m-3}\,dx}{x^{\displaystyle3}+a^{\displaystyle3}}\)
- 12.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle n}(x^{\displaystyle3}+a^{\displaystyle3})}=\displaystyle \frac{-1}{a^{\displaystyle3}(n-1)x^{\displaystyle n-1}}-\displaystyle \frac{1}{a^{\displaystyle3}}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle n-3}(x^{\displaystyle3}+a^{\displaystyle3})}\)
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