Matematica

Con $\small x^2-a^2$

$x^2>a^2\quad \text{kabul edecektir}$

1.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x-a}{x+a}\right)\;\; \text{veya}\;\;-\displaystyle \frac{1}{a}coth^ {\displaystyle-1}\displaystyle \frac{x}{a}$

2.: $\displaystyle \int\displaystyle \frac{x\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{1}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})$

3.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=x+\displaystyle \frac{a}{2}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$

4.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{x^{\displaystyle2}-a^{\displaystyle2}}=\displaystyle \frac{x^{\displaystyle2}}{2}+\displaystyle \frac{a^{\displaystyle2}}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})$

5.: $\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{2a^{\displaystyle2}}\ln\left(\displaystyle \frac{x^{\displaystyle2}-a^{\displaystyle2}}{x^{\displaystyle2}}\right)$

6.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{a^{\displaystyle2}x}+\displaystyle \frac{1}{2a^{\displaystyle3}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$

7.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}-a^{\displaystyle2})}=\displaystyle \frac{1}{2a^{\displaystyle2}x^{\displaystyle2}}-\displaystyle \frac{1}{2a^{\displaystyle4}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

8.: $\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-x}{2a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}-\displaystyle \frac{1}{4a^{\displaystyle3}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$

9.: $\displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-1}{2(x^{\displaystyle2}-a^{\displaystyle2})}$

10.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-x}{2(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{4a}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$

11.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-a^{\displaystyle2}}{2(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{2}\ln(x^{\displaystyle2}-a^{\displaystyle2})$

12.: $\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=\displaystyle \frac{-1}{2a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{2a^{\displaystyle4}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

13.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=-\displaystyle \frac{1}{a^{\displaystyle4}x}-\displaystyle \frac{x}{2a^{\displaystyle4}(x^{\displaystyle2}-a^{\displaystyle2})}-\displaystyle \frac{3}{4a^{\displaystyle5}}\ln\left(\displaystyle \frac{x-a}{x+a}\right)$

14.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle2}}=-\displaystyle \frac{1}{2a^{\displaystyle4}x^{\displaystyle2}}-\displaystyle \frac{1}{2a^{\displaystyle4}(x^{\displaystyle2}-a^{\displaystyle2})}+\displaystyle \frac{1}{a^{\displaystyle6}}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

15.: $\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-x}{2(n-1)a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}-\displaystyle \frac{2n-3}{(2n-2)a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}$

16.: $\displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-1}{2(n-1)(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}$

17.: $\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{-1}{2(n-1)a^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}-\displaystyle \frac{1}{a^{\displaystyle2}}\d$

18.: $\displaystyle \int\displaystyle \frac{x^{\displaystyle m}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \int\displaystyle \frac{x^{\displaystyle m-2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}+a^{\displaystyle2}\displaystyle \int\displaystyle \frac{x^{\displaystyle m-2}\,dx}{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}$

19.: $\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}=\displaystyle \frac{1}{a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m-2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n}}-\displaystyle \frac{1}{a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle n-1}}$