 Matematica
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Con $\small \sqrt{ax+b}$ e $\small \sqrt{px+q}$
- 1.
- $\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{(ax+b)(px+q)}}=\left\{\begin{array}{l}
\displaystyle \frac{2}{\displaystyle \sqrt{ap}}\ln\left(\displaystyle \sqrt{a(px+q)}+\displaystyle \sqrt{p(ax+b)}\right)\\ \\ \displaystyle \frac{2}{\displaystyle \sqrt{-ap}}\tan^{\displaystyle-1}\displaystyle \sqrt{\displaystyle \frac{-p(ax+b)}{a(px+q)}} \end{array} \right.$
- 2.
- $\displaystyle \int\displaystyle \frac{x\,dx}{\displaystyle \sqrt{(ax+b)(px+q)}}=\displaystyle \frac{\displaystyle \sqrt{(ax+b)(px+q)}}{ap}-\displaystyle \frac{bp+aq}{2ap}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{(ax+b)(px+q)}}$
- 3.
- $\begin{array}{lcl}
\displaystyle \int\displaystyle \sqrt{(ax+b)(px+q)}\,dx&=&\displaystyle \frac{2apx+bp+aq}{4ap}\displaystyle \sqrt{(ax+b)(px+q)}\\ &&\\ && -\displaystyle \frac{(bp-aq)^{\displaystyle2}}{8ap}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{(ax+b)(px+q)}}\end{array}$
- 4.
- $\displaystyle \int\displaystyle \sqrt{\displaystyle \frac{px+q}{ax+b}}\,dx=\displaystyle \frac{\displaystyle \sqrt{(ax+b)(px+q)}}{a}+\displaystyle \frac{aq-bp}{2a}\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{(ax+b)(px+q)}}$
- 5.
- $\displaystyle \int\displaystyle \frac{dx}{(px+q)\displaystyle \sqrt{(ax+b)(px+q)}}=\displaystyle \frac{2\displaystyle \sqrt{ax+b}}{(aq-bp)\displaystyle \sqrt{px+q}}$
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