 Matematik
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$\sqrt{ax^2+bx+c}$ içeren integraller
- 1.
- \(\small \int\displaystyle \frac{dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}=\left\{\begin{array}{l}
\displaystyle \frac{1}{\displaystyle \sqrt{a}}\ln\left(2\displaystyle \sqrt{a}\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}+2ax+b\right)\\ -\displaystyle \frac{1}{\displaystyle \sqrt{-a}}\sin^{\displaystyle-1}\left(\displaystyle \frac{2ax+b}{\displaystyle \sqrt{b^{\displaystyle2}-4ac}}\right)\\ or\\ \displaystyle \frac{1}{\displaystyle \sqrt{a}}\sinh^{\displaystyle-1}\left(\displaystyle \frac{2ax+b}{\displaystyle \sqrt{4ac-b^{\displaystyle2}}}\right)
\end{array}
\right.\)
- 2.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}=\displaystyle \frac{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}{a}\;-\;\displaystyle \frac{b}{2a}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\)
- 3.
- \(\small \begin{array}{lcl} \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}&=&\displaystyle \frac{2ax-3b}{4a^{\displaystyle2}}\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}\;\\ \\&&+\;\displaystyle \frac{3b^{\displaystyle2}-4ac}{8a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}} \end{array}\)
- 4.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}=\left\{\begin{array}{l}
-\displaystyle \frac{1}{\displaystyle \sqrt{c}}\ln\left(\displaystyle \frac{2\displaystyle \sqrt{c}\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}+bx+2c}{x}\right)\\ \\ \displaystyle \frac{1}{\displaystyle \sqrt{-c}}\sin^{\displaystyle-1}\left(\displaystyle \frac{bx+2c}{\left|x\right|\displaystyle \sqrt{b^{\displaystyle2}-4ac}}\right)\\ \mbox{or}\\ -\displaystyle \frac{1}{\displaystyle \sqrt{c}}\sinh^{\displaystyle-1}\left(\displaystyle \frac{bx+2c}{\left|x\right|\displaystyle \sqrt{4ac-b^{\displaystyle2}}}\right)
\end{array}
\right.\)
- 5.
- \(\small \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}=-\displaystyle \frac{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}{cx}\;-\;\displaystyle \frac{b}{2c}\displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\)
- 6.
- \(\small \begin{array}{lcl} \displaystyle \int\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}\,dx&=&\displaystyle \frac{(2ax+b)\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}{4a}\;\\&&+\;\displaystyle \frac{4ac-b^{\displaystyle2}}{8a}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}} \end{array}\)
- 7.
- \(\small \small \displaystyle
\begin{array}{lclcl}
\displaystyle \int x\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}\,dx & = & \displaystyle \frac{(ax^{\displaystyle2}+bx+c)^{\displaystyle3/2}}{3a}\;-\;\displaystyle \frac{b(2ax+b)}{8a^{\displaystyle2}}\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}\\&& - \displaystyle \frac{b(4ac-b^{\displaystyle2})}{16a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}
\end{array}\)
- 8.
- \(\small \begin{array}{lcl} \displaystyle \int x^{\displaystyle2}\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}\,dx & =&\displaystyle \frac{6ax-5b}{24a^{\displaystyle2}}\left(ax^{\displaystyle2}+bx+c\right)^{\displaystyle3/2}\;\\ && \\&&+\;\displaystyle \frac{5b^{\displaystyle2}-4ac}{16a^{\displaystyle2}}\displaystyle \int\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}\,dx \end{array}\)
- 9.
- \(\small \begin{array}{lcl} \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}{x}\,dx&=&\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}\;+\;\displaystyle \frac{b}{2}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\;\\&&+\;c\displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}} \end{array}\)
- 10.
- \(\small \begin{array}{lcl} \displaystyle \int\displaystyle \frac{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}{x^{\displaystyle2}}\,dx &=& -\displaystyle \frac{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}{x}\;+\;a\displaystyle \int\displaystyle \frac{dx}{\displaystyle{ax^{\displaystyle2}+bx+c}}\;\\ \\&&+\;\displaystyle \frac{b}{2}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}} \end{array}\)
- 11.
- \(\small \displaystyle \int\displaystyle \frac{dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle3/2}}=\displaystyle \frac{2(2ax+b)}{(4ac-b^{\displaystyle2})\,\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\)
- 12.
- \(\small \displaystyle \int\displaystyle \frac{x\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle3/2}}=\displaystyle \frac{2(bx+2c)}{(b^{\displaystyle2}-4ac)\,\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\)
- 13.
- \(\small \small \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle3/2}}=\displaystyle \frac{(2b^{\displaystyle2}-4ac)x+2bc}{a(4ac-b^{\displaystyle2})\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}+\displaystyle \frac{1}{a}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\)
- 14.
- \(\small \small \begin{array}{lcl}
\displaystyle \int\displaystyle \frac{dx}{x(ax^{\displaystyle2}+bx+c)^{\displaystyle3/2}}&=&\displaystyle \frac{1}{c\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\,+\,\displaystyle \frac{1}{c}\displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\\&&\\&&-\;\displaystyle \frac{b}{2c}\displaystyle \int\displaystyle \frac{dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle3/2}}\end{array}\)
- 15.
- \(\small \displaystyle
\begin{array}{lcccl}
\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(ax^{\displaystyle2}+bx+c)^{\displaystyle3/2}} =&& -\displaystyle \frac{ax^{\displaystyle2}+2bx+c}{c^{\displaystyle2}x\displaystyle \sqrt{ax^{\displaystyle2}+bx+c}}\,\\ \\&&+\,\displaystyle \frac{b^{\displaystyle2}-2ac}{2c^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle3/2}}\\\ &&
\end{array}\)
- 16.
- \(\small \displaystyle
\begin{array}{lcl}
\displaystyle \int(ax^{\displaystyle2}+bx+c)^{\displaystyle n+1/2}\,dx = \displaystyle \frac{(2ax+b)(ax^{\displaystyle2}+bx+c)^{\displaystyle n+1/2}}{4a(n+1)}\\\quad\quad\quad\quad\quad\quad
\end{array}\)
- 17.
- \(\small \begin{array}{llll}
\displaystyle \int x(ax^{\displaystyle2}+bx+c)^{\displaystyle n+1/2}\,dx=\displaystyle \frac{(ax^{\displaystyle2}+bx+c)^{\displaystyle n+3/2}}{a(2n+3)}\,\\ -\,\displaystyle \frac{b}{2a}\displaystyle \int(ax^{\displaystyle2}+bx+c)^{\displaystyle n+1/2}\,dx
\end{array}\)
- 18.
- \(\small \small \displaystyle
\begin{array}{lcl}
\displaystyle \int\displaystyle \frac{dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle n+1/2}} & = & \displaystyle \frac{2(2ax+b)}{(2n-1)(4ac-b^{\displaystyle2})(ax^{\displaystyle2}+bx+c)^{\displaystyle n-1/2}}\\
\end{array}\)
- 19.
- \(\small \displaystyle
\begin{array}{ll}
\displaystyle \int\displaystyle \frac{dx}{x(ax^{\displaystyle2}+bx+c)^{\displaystyle n+1/2}} = &\displaystyle \frac{1}{(2n-1)c(ax^{\displaystyle2}+bx+c)^{\displaystyle n-1/2}}\\&\\
\end{array}\)
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