 Matematik
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\(a^2-x^2\) içeren integraller
- \(\small a^2>x^2\quad \text{kabul edecektir.}\)
- 1.
- \(\small \displaystyle \int\displaystyle \frac{dx}{a^{\displaystyle2}-x^{\displaystyle2}}=\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{a+x}{a-x}\right)\;\;\text{veya}\;\;\;\;\displaystyle \frac{1}{a}\tanh^{\displaystyle-1}\displaystyle \frac{x}{a}\)
- 2.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{a^{\displaystyle2}-x^{\displaystyle2}}=-\displaystyle \frac{1}{2}\ln(a^{\displaystyle2}-x^{\displaystyle2})\)
- 3.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{a^{\displaystyle2}-x^{\displaystyle2}}=-x+\displaystyle \frac{a}{2}\ln\left(\displaystyle \frac{a+x}{a-x}\right)\)
- 4.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{a^{\displaystyle2}-x^{\displaystyle2}}=-\displaystyle \frac{x^{\displaystyle2}}{2}-\displaystyle \frac{a^{\displaystyle2}}{2}\ln(a^{\displaystyle2}-x^{\displaystyle2})\)
- 5.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle2})}=\displaystyle \frac{1}{2a^{\displaystyle2}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)\)
- 6.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(a^{\displaystyle2}-x^{\displaystyle2})}=-\displaystyle \frac{1}{a^{\displaystyle2}x}+\displaystyle \frac{1}{2a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+x}{a-x}\right)\)
- 7.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}=-\displaystyle \frac{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{2a^{\displaystyle2}x^{\displaystyle2}}\;-\;\displaystyle \frac{1}{2a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x}\right)\)
- 8.
- \(\small \displaystyle \int\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}\,dx=\displaystyle \frac{x\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{2}\;+\;\displaystyle \frac{a^{\displaystyle2}}{2}\sin^{\displaystyle-1}\displaystyle \frac{x}{a}\)
- 9.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{1}{2(a^{\displaystyle2}-x^{\displaystyle2})}\)
- 10.
- \(\small \displaystyle \int x^{\displaystyle2}\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}\,dx=-\displaystyle \frac{x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle3/2}}{4}\;+\;\displaystyle \frac{a^{\displaystyle2}x\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{8}\;+\;\displaystyle \frac{a^{\displaystyle4}}{8}\sin^{\displaystyle-1}\displaystyle \frac{x}{a}\)
- 11.
- \(\small \displaystyle \int x^{\displaystyle3}\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}\,dx=\displaystyle \frac{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle5/2}}{5}\;-\;\displaystyle \frac{a^{\displaystyle2}(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}{3}\)
- 12.
- \(\small \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x}\,dx=\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}-a\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x}\right)\)
- 13.
- \(\small \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x^{\displaystyle2}}\,dx=-\displaystyle \frac{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x}-\sin^{\displaystyle-1}\displaystyle \frac{x}{a}\)
- 14.
- \(\small \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x^{\displaystyle3}}\,dx=-\displaystyle \frac{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{2x^{\displaystyle2}}\;+\;\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x}\right)\)
- 15.
- \(\small \displaystyle \int\displaystyle \frac{dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{x}{a^{\displaystyle2}\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}\)
- 16.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{1}{2(n-1)(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}\)
- 17.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{x}{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}-\sin^{\displaystyle-1}\displaystyle \frac{x}{a}\)
- 18.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}+\displaystyle \frac{a^{\displaystyle2}}{\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}\)
- 19.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle3/2}}=\displaystyle \frac{1}{a^{\displaystyle2}\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}-\displaystyle \frac{1}{a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^{\displaystyle2}-x^{\displaystyle2}}}{x}\right)\)
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