 Matematik
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\(x^2+a^2\) içeren integraller
- 1.
- \(\small \displaystyle\int \displaystyle\frac{dx}{x^{2}+a^{2}}=\displaystyle\frac{1}{a}\displaystyle\tan^{-1} \displaystyle\frac{x}{a}\)
- 2.
- \(\small \displaystyle\int \displaystyle\frac{x\,dx}{x^{2}+a^{2}}=\displaystyle\frac{1}{2}\displaystyle\ln(x^{2}+a^{2})\)
- 3.
- \(\small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{x^{\displaystyle2}+a^{\displaystyle2}}=x-a\tan^{\displaystyle-1}\displaystyle \frac{x}{a}\)
- 4.
- \(\small \displaystyle\int \displaystyle\frac{x^{3}\,dx}{x^{2}+a^{2}}= \displaystyle\frac{x^{2}}{2}- \displaystyle\frac{a^{2}}{2}\displaystyle\ln(x^{2}+a^{2})\)
- 5.
- \(\small \displaystyle\int \displaystyle\frac{dx}{x(x^{2}+a^{2})}= \displaystyle\frac{1}{2a^{2}}\displaystyle\ln\left( \displaystyle\frac{x^{2}}{x^{2}+a^{2}}\right)\)
- 6.
- \(\small \displaystyle\int \displaystyle\frac{dx}{x^{2}(x^{2}+a^{2})}=- \displaystyle\frac{1}{a^{2}x}- \displaystyle\frac{1}{a^{3}}\displaystyle\tan^{-1} \displaystyle\frac{x}{a}\)
- 7.
- \(\small \displaystyle\int \displaystyle\frac{dx}{x^{3}(x^{2}+a^{2})}=- \displaystyle\frac{1}{2a^{2}x^{2}}- \displaystyle\frac{1}{2a^{4}}\displaystyle\ln\left( \displaystyle\frac{x^{2}}{x^{2}+a^{2}}\right)\)
- 8.
- \(\small \displaystyle\int \displaystyle\frac{dx}{(x^{2}+a^{2})^{2}}=\displaystyle\frac{x}{2a^{2}(x^{2}+a^{2})}+ \displaystyle\frac{1}{2a^{3}}\displaystyle\tan^{-1} \displaystyle\frac{x}{a}\)
- 9.
- \(\small \displaystyle\int \displaystyle\frac{x\,dx}{(x^{2}+a^{2})^{2}}= \displaystyle\frac{-1}{2(x^{2}+a^{2})}\)
- 10.
- \(\small \displaystyle\int \displaystyle\frac{x^{2}\,dx}{(x^{2}+a^{2})^{2}}= \displaystyle\frac{-x}{2(x^{2}+a^{2})}+\displaystyle\frac{1}{2a}\displaystyle\tan^{-1} \displaystyle\frac{x}{a}\)
- 11.
- \(\small \displaystyle\int \displaystyle\frac{x^{3}\,dx}{(x^{2}+a^{2})^{2}}= \displaystyle\frac{a^{2}}{2(x^{2}+a^{2})}+ \displaystyle\frac{1}{2}\displaystyle\ln(x^{2}+a^{2})\)
- 12.
- \(\small \displaystyle\int\displaystyle\frac{dx}{x(x^{2}+a^{2})^{2}}=\displaystyle\frac{1}{2a^{2}(x^{2}+a^{2})}+\displaystyle\frac{1}{2a^{4}}\displaystyle\ln\left( \displaystyle\frac{x^{2}}{x^{2}+a^{2}}\right)\)
- 13.
- \(\small \displaystyle\int \displaystyle\frac{dx}{x^{2}(x^{2}+a^{2})^{2}}=- \displaystyle\frac{1}{a^{4}x}-\displaystyle\frac{x}{2a^{4}(x^{2}+a^{2})}-\displaystyle\frac{3}{2a^{5}}\displaystyle\tan^{-1}\displaystyle\frac{x}{a}\)
- 14.
- \(\small \displaystyle\int\displaystyle\frac{dx}{x^{3}(x^{2}+a^{2})^{2}}=-\displaystyle\frac{1}{2a^{4}x^{2}}-\displaystyle\frac{1}{2a^{4}(x^{2}+a^{2})}- \displaystyle\frac{1}{a^{6}}\ln\left( \displaystyle\frac{x^{2}}{x^{2}+a^{2}}\right)\)
- 15.
- \(\small \displaystyle\int\displaystyle\frac{dx}{(x^{2}+a^{2})^{ n}}= \displaystyle\frac{x}{2(n-1)a^{2}(x^{2}+a^{2})^{n-1}}+\displaystyle\frac{2n-3}{(2n-2)a^{2}} \displaystyle\int \displaystyle\frac{dx}{(x^{2}+a^{2})^{n-1}}\)
- 16.
- \(\small \displaystyle\int\displaystyle\frac{x\,dx}{(x^{2}+a^{2})^{n}}=\displaystyle\frac{-1}{2(n-1)(x^{2}+a^{2})^{n-1}}\)
- 17.
- \(\small \displaystyle\int\displaystyle\frac{dx}{x(x^{2}+a^{2})^{n}}=\displaystyle\frac{1}{2(n-1)a^{2}(x^{2}+a^{2})^{n-1}}+\displaystyle\frac{1}{a^{2}}\displaystyle\int\displaystyle\frac{dx}{x(x^{2}+a^{2})^{n-1}}\)
- 18.
- \(\small \displaystyle\int\displaystyle\frac{x^{m}\,dx}{(x^{2}+a^{2})^{n}}=\displaystyle\int\displaystyle\frac{x^{m-2}\,dx}{(x^{2}+a^{2})^{n-1}}-a^{2} \displaystyle\int\displaystyle\frac{x^{m-2}\,dx}{(x^{2}+a^{2})^{n}}\)
- 19.
- \(\small \displaystyle\int\displaystyle\frac{dx}{x^{m}(x^{2}+a^{2})^{n}}=\displaystyle\frac{1}{a^{2}}\displaystyle\int\displaystyle\frac{dx}{x^{m}(x^{2}+a^{2})^{n-1}}- \displaystyle\frac{1}{a^{2}}\displaystyle\int\displaystyle\frac{dx}{x^{m-2}(x^{2}+a^{2})^{n}}\)
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