 Matematik
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\(x^2-a^2\) içeren integraller
- \(\small x^2>a^2\quad \text{kabul edecektir}\)
- 1.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\;\; \text{veya}\;\;-\displaystyle \frac{1}{a}coth^ {\displaystyle-1}\displaystyle \frac{x}{a}\)
- 2.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{1}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})\)
- 3.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=x+\displaystyle \frac{a}{2}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\)
- 4.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{x^{\displaystyle2}}{2}+\displaystyle \frac{a^{\displaystyle2}}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})\)
- 5.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{2a^{\displaystyle2}}\ln\left(\displaystyle \frac{x^{\displaystyle2}-a^{\displaystyle2}}{x^{\displaystyle2}}\right)\)
- 6.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{a^{\displaystyle2}x}+\displaystyle \frac{1}{2a^{\displaystyle3}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\)
- 7.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{2a^{\displaystyle2}x^{\displaystyle2}}-\displaystyle \frac{1}{2a^{\displaystyle4}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)\)
- 8.
- \(\small \displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-x}{2a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}-\displaystyle \frac{1}{4a^{\displaystyle3}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\)
- 9.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-1}{2(x^{\displaystyle2}-a^{\displaystyle2})}\)
- 10.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-x}{2(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{4a}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\)
- 11.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-a^{\displaystyle2}}{2(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})\)
- 12.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-1}{2a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{2a^{\displaystyle4}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)\)
- 13.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=-\displaystyle \frac{1}{a^{\displaystyle4}x}-\displaystyle \frac{x}{2a^{\displaystyle4}(x^{\displaystyle2}-a^{\displaystyle2})}-\displaystyle \frac{3}{4a^{\displaystyle5}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\)
- 14.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=-\displaystyle \frac{1}{2a^{\displaystyle4}x^{\displaystyle2}}-\displaystyle \frac{1}{2a^{\displaystyle4}(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{a^{\displaystyle6}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)\)
- 15.
- \(\small \displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-x}{2(n-1)a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}-\displaystyle \frac{2n-3}{(2n-2)a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}\)
- 16.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-1}{2(n-1)(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}\)
- 17.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-1}{2(n-1)a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}-\displaystyle \frac{1}{a^{\displaystyle2}}\d\)
- 18.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle m}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \int\displaystyle \frac{x^{\displaystyle m-2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}+a^{\displaystyle2}\displaystyle \int\displaystyle \frac{x^{\displaystyle m-2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}\)
- 19.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{1}{a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m-2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}-\displaystyle \frac{1}{a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}\)
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