 Matematik
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\(\sqrt{x^2+a^2}\) içeren integraller
- 1.
- \(\small \displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)\;\;\text{veya}\;\;sinh^{\displaystyle-1}\displaystyle \frac{x}{a}\)
- 2.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\)
- 3.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=\displaystyle \frac{x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{2}-\displaystyle \frac{a^{\displaystyle2}}{2}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)\)
- 4.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{3}-a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\)
- 5.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=-\displaystyle \frac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)\)
- 6.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=-\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{a^{\displaystyle2}x}\)
- 7.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}=-\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{2a^{\displaystyle2}x^{\displaystyle2}}+\displaystyle \frac{1}{2a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)\)
- 8.
- \(\small \displaystyle \int\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\,dx=\displaystyle \frac{x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{2}+\displaystyle \frac{a^{\displaystyle2}}{2}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)\)
- 9.
- \(\small \displaystyle \int x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{3}\)
- 10.
- \(\small \small \displaystyle \int x^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\,dx=\displaystyle \frac{x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{4}\;-\;\displaystyle \frac{a^{\displaystyle2}x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{8}\;-\;\displaystyle \frac{a^{\displaystyle4}}{8}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)\)
- 11.
- \(\small \displaystyle \int x^{\displaystyle3}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle5/2}}{5}-\displaystyle \frac{a^{\displaystyle2}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{3}\)
- 12.
- \(\small \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\,dx=\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}-a\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)\)
- 13.
- \(\small \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x^{\displaystyle2}}\,dx= -\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}+\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)\)
- 14.
- \(\small \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x^{\displaystyle3}}\,dx=-\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{2x^{\displaystyle2}}\;-\;\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)\)
- 15.
- \(\small \displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{x}{a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}\)
- 16.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{-1}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}\)
- 17.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{-x}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}+\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)\)
- 18.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}+\displaystyle \frac{a^{\displaystyle2}}{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}\)
- 19.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{1}{a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}-\displaystyle \frac{1}{a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)\)
- 20.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=-\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{a^{\displaystyle4}x}\;-\;\displaystyle \frac{x}{a^{\displaystyle4}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}\)
- 21.
- \(\small \small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{-1}{2a^{\displaystyle2}x^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}\;-\;\displaystyle \frac{3}{2a^{\displaystyle4}\displaystyle \sqrt{x^{\displaystyle2}\;+\;a^{\displaystyle2}}}+\displaystyle \frac{3}{2a^{\displaystyle5}}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)\)
- 22.
- \(\small \small \displaystyle \int(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}\,dx=\displaystyle \frac{x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{4}\;+\;\displaystyle \frac{3a^{\displaystyle2}x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{8}\;+\;\displaystyle \frac{3}{8}a^{\displaystyle4}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)\)
- 23.
- \(\small \displaystyle \int x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle5/2}}{5}\)
- 24.
- \(\small \begin{array}{lcl}
\displaystyle \int x^{\displaystyle2}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}\,dx&=&\displaystyle \frac{x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle5/2}}{6}\; -\;\displaystyle \frac{a^{\displaystyle2}x(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{24}\;\\ \\ && -\;\displaystyle \frac{a^{\displaystyle4}x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{16}\;-\;\displaystyle \frac{a^{\displaystyle6}}{16}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)
\end{array}\)
- 25.
- \(\small \displaystyle \int x^{\displaystyle3}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle7/2}}{7}\;-\;\displaystyle \frac{a^{\displaystyle2}(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle5/2}}{5}\)
- 26.
- \(\small \small \displaystyle \int\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{x}\,dx=\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{3}\;+\;a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\;-\;a^{\displaystyle3}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)\)
- 27.
- \(\small \small \displaystyle \int\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{x^{\displaystyle2}}\,dx=-\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{x}\;+\;\displaystyle \frac{3x\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{2}\;+\;\displaystyle \frac{3}{2}a^{\displaystyle2}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}\right)\)
- 28.
- \(\small \displaystyle \int\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{x^{\displaystyle3}}\,dx=-\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{\displaystyle3/2}}{2x^{\displaystyle2}}\;+\;\displaystyle \frac{3}{2}\d\)
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