 Matematica
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Con $\small ax^2+bx+c$
- 1.
- $\displaystyle \int\displaystyle \frac{dx}{ax^{\displaystyle2}+bx+c}=\left\{\begin{array}{l}
\displaystyle \frac{2}{\displaystyle \sqrt{4ac-b^{\displaystyle2}}}\tan^{\displaystyle-1}\displaystyle \frac{2ax+b}{\displaystyle \sqrt{4ac-b^{\displaystyle2}}} \\ \displaystyle \frac{1}{\displaystyle \sqrt{b^{\displaystyle2}-4ac}}\ln\left(\displaystyle \frac{2ax+b-\displaystyle \sqrt{b^{\displaystyle2}-4ac}}{2ax+b+\displaystyle \sqrt{b^{\displaystyle2}-4ac}}\right)
\end{array} \right.$
- 2.
- $\displaystyle \int\displaystyle \frac{x\,dx}{ax^{\displaystyle2}+bx+c}=\displaystyle \frac{1}{2a}\ln\left(ax^{\displaystyle2}+bx+c\right)\;-\;\displaystyle \frac{b}{2a}\displaystyle \int\displaystyle \frac{dx}{ax^{\displaystyle2}+bx+c}$
- 3.
- $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{ax^{\displaystyle2}+bx+c}&=&\displaystyle \frac{x}{a}\;-\;\displaystyle \frac{b}{2a^{\displaystyle2}}\ln\left(ax^{\displaystyle2}+bx+c\right)\;\\\\&&+\;\displaystyle \frac{b^{\displaystyle2}-2ac}{2a^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{ax^{\displaystyle2}+bx+c} \end{array}$
- 4.
- $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x^{\displaystyle m}}{ax^{\displaystyle2}+bx+c}&=&\displaystyle \frac{x^{\displaystyle m-1}}{(m-1)a}\;-\;\displaystyle \frac{c}{a}\displaystyle \int\displaystyle \frac{x^{\displaystyle m-2}\,dx}{ax^{\displaystyle2}+bx+c}\;\\\\&&-\;\displaystyle \frac{b}{a}\displaystyle \int\displaystyle \frac{x^{\displaystyle m-1}\,dx}{ax^{\displaystyle2}+bx+c} \end{array}$
- 5.
- $\displaystyle \int\displaystyle \frac{dx}{x(ax^{\displaystyle2}+bx+c)}=\displaystyle \frac{1}{2c}\ln\left(\displaystyle \frac{x^{\displaystyle2}}{ax^{\displaystyle2}+bx+c}\right)\;-\;\displaystyle \frac{b}{2c}\displaystyle \int\displaystyle \frac{dx}{ax^{\displaystyle2}+bx+c}$$
- 6.
- $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(ax^{\displaystyle2}+bx+c)}&=&\displaystyle \frac{b}{2c^{\displaystyle2}}\ln\left(\displaystyle \frac{ax^{\displaystyle2}+bx+c}{x^{\displaystyle2}}\right)\;-\;\displaystyle \frac{1}{cx}\;\\&&+\;\displaystyle \frac{b^{\displaystyle2}-2ax}{2c^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx} {ax^{\displaystyle2}+bx+c} \end{array}$
- 7.
- $\begin{array}{ll}
\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle n}(ax^{\displaystyle2}+bx+c)}= -\displaystyle \frac{1}{(n-1)cx^{\displaystyle n-1}}\;-\;\displaystyle \frac{b}{c}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle n-1}(ax^{\displaystyle2}+bx+c)}\;\\ -\;\displaystyle \frac{a}{c}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle n-2}(ax^{\displaystyle2}+bx+c)}
\end{array}$
- 8.
- $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle2}}&=&\displaystyle \frac{2ax+b}{(4ac-b^{\displaystyle2})(ax^{\displaystyle2}+bx+c)}\;\\\\&&+\;\displaystyle \frac{2a}{4ac-b^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{ax^{\displaystyle2}+bx+c} \end{array}$
- 9.
- $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle2}}&=&-\displaystyle \frac{bx+2c}{(4ac-b^{\displaystyle2})(ax^{\displaystyle2}+bx+c)}\;\\\\&&-\;\displaystyle \frac{b}{4ac-b^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{ax^{\displaystyle2}+bx+c} \end{array}$
- 10.
- $\begin{array}{lcl} \displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle2}}&=&\displaystyle \frac{(b^{\displaystyle2}-2ac)x+bc}{a(4ac-b^{\displaystyle2})(ax^{\displaystyle2}+bx+c)}\;\\\\&&+\;\displaystyle \frac{2c}{4ac-b^{\displaystyle2}}\displaystyle \int\displaystyle \frac{dx}{ax^{\displaystyle2}+bx+c} \end{array}$
- 11.
- $\begin{array}{ll}
\displaystyle \int\displaystyle \frac{x^{\displaystyle m}\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle n}} = -\displaystyle \frac{x^{\displaystyle m-1}}{(2n-m-1)a(ax^{\displaystyle2}+bx+c)^{\displaystyle n-1}}\;\\ + \;\displaystyle \frac{(m-1)c}{(2n-m-1)a}\displaystyle \int\displaystyle \frac{x^{\displaystyle m-2}\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle n}}
\end{array}$
- 12.
- $\begin{array}{ll}
\displaystyle \int\displaystyle \frac{x^{\displaystyle2n-1}\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle n}}=\displaystyle \frac{1}{a}\displaystyle \int\displaystyle \frac{x^{\displaystyle2n-3}\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle n-1}}\;\\\\\quad\quad\quad\quad\quad\quad\quad\quad\quad-\;\displaystyle \frac{c}{a}\displaystyle \int\displaystyle \frac{x^{\displaystyle2n-3}\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle n}}\;\\\quad\quad\quad\quad\quad\quad\quad\quad\quad -\;\displaystyle \frac{b}{a}\displaystyle \int\displaystyle \frac{x^{\displaystyle2n-2}\,dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle n}}
\end{array}$
- 13.
- $\begin{array}{ll}
\displaystyle \int\displaystyle \frac{dx}{x(ax^{\displaystyle2}+bx+c)^{\displaystyle2}}=\displaystyle \frac{1}{2c(ax^{\displaystyle2}+bx+c)}\;\\\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\;\displaystyle \frac{b}{2c}\displaystyle \int\displaystyle \frac{dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle2}}\;\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad +\;\displaystyle \frac{1}{c}\displaystyle \int\displaystyle \frac{dx}{x(ax^{\displaystyle2}+bx+c)}
\end{array}$
- 14.
- $\begin{array}{ll}
\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(ax^{\displaystyle2}+bx+c)^{\displaystyle2}}=-\displaystyle \frac{1}{cx(ax^{\displaystyle2}+bx+c)}\;\\\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad-\;\displaystyle \frac{3a}{c}\displaystyle \int\displaystyle \frac{dx}{(ax^{\displaystyle2}+bx+c)^{\displaystyle2}}\;\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad -\;\displaystyle \frac{2b}{c}\displaystyle \int\displaystyle \frac{dx}{x(ax^{\displaystyle2}+bx+c)^{\displaystyle2}}
\end{array}$
- 15.
- $\begin{array}{lcl}
\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(ax^{\displaystyle2}+bx+c)^{\displaystyle n}} = -\displaystyle \frac{1}{(m-1)cx^{\displaystyle m-1}(ax^{\displaystyle2}+bx+c)^{\displaystyle n-1}}\;\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad -\;\displaystyle \frac{(m+2n-3))a}{(m-1)c}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m-2}(ax^{\displaystyle2}+bx+c)^{\displaystyle n}}\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad -\displaystyle \frac{(m+n-2)b}{(m-1)c}\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m-1}(ax^{\displaystyle2}+bx+c)^{\displaystyle n}}
\end{array}$
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