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Matematica

Con $\small \sqrt{ax+b}$

1.: $\displaystyle\int{\displaystyle\frac{dx}{\displaystyle\sqrt{ax+b}}}=\displaystyle\frac{2\displaystyle\sqrt{ax+b}}{a}$

2.: $\displaystyle\int{\displaystyle\frac{xdx}{\displaystyle\sqrt{ax+b}}}=\displaystyle\frac{2(ax-2b)}{3a^2}\displaystyle\sqrt{ax+b}$

3.: $\displaystyle\int{\displaystyle\frac{x^2dx}{\displaystyle\sqrt{ax+b}}}=\displaystyle\frac{2(3a^2x^2-4abx+8b^2)}{15a^3}\displaystyle\sqrt{ax+b}$

4.: $\displaystyle\int{\displaystyle\frac{dx}{x(ax+b)}}= \left\{\begin{matrix} \displaystyle\frac{1}{\displaystyle\sqrt{b}}\ln{\left( \frac{\displaystyle\sqrt{ax+b}-\displaystyle\sqrt{b}}{\displaystyle\sqrt{ax+b}+\displaystyle\sqrt{b}} \right)}\\ \displaystyle\frac{2}{\displaystyle\sqrt{-b}}\displaystyle\tan^{-1}\displaystyle\sqrt{\displaystyle\frac{ax+b}{-b}} \end{matrix}\right.$

5.: $\displaystyle\int{\displaystyle\frac{dx}{x^2\displaystyle\sqrt{ax+b}}}= -\displaystyle\frac{\displaystyle\sqrt{ax+b}}{bx}-\displaystyle\frac{a}{2b}\displaystyle\int{\displaystyle\frac{dx}{x\displaystyle\sqrt{ax+b}}}$

6.: $\displaystyle\int{\displaystyle\sqrt{ax+b}dx}=\displaystyle\frac{2\displaystyle\sqrt{(ax+b)^3}}{3a}$

7.: $\displaystyle\int{x\displaystyle\sqrt{ax+b}dx}=\displaystyle\frac{2(3ax-2b)}{15a^2}\displaystyle\sqrt{(ax+b)^3}$

8.: $\displaystyle\int{x^2\displaystyle\sqrt{ax+b}dx}=\displaystyle\frac{2(15a^2x^2-12abx+8b^2)}{105a^3}\displaystyle\sqrt{(ax+b)^3}$

9.: $\displaystyle\int{\displaystyle\frac{\displaystyle\sqrt{ax+b}}{x}dx}=2\displaystyle\sqrt{ax+b}+b\displaystyle\int\displaystyle\frac{dx}{x\displaystyle\sqrt{ax+b}}$

10.: $\displaystyle\int{\displaystyle\frac{\displaystyle\sqrt{ax+b}}{x^2}dx}=-\displaystyle\frac{\displaystyle\sqrt{ax+b}}{x}+\displaystyle\frac{a}{2}\displaystyle\int{\displaystyle\frac{dx}{x\displaystyle\sqrt{ax+b}}}$

11.: $\displaystyle\int{\displaystyle\frac{x^m}{\displaystyle\sqrt{ax+b}}dx}=\displaystyle\frac{2x^m\displaystyle\sqrt{ax+b}}{(2m+1)a}-\displaystyle\frac{2mb}{(2m+1)a}\displaystyle\int{\displaystyle\frac{x^{m-1}}{\displaystyle\sqrt{ax+b}}dx}$

12.: $\displaystyle\int{\displaystyle\frac{dx}{x^m\displaystyle\sqrt{ax+b}}}=-\displaystyle\frac{\displaystyle\sqrt{ax+b}}{(m-1)bx^{m-1}}-\displaystyle\frac{(2m-3)a}{(2m-2)b}\displaystyle\int{\displaystyle\frac{dx}{x^{m-1}\displaystyle\sqrt{ax+b}}}$

13.: $\begin{align*} \displaystyle\int{x^m\displaystyle\sqrt{ax+b}dx}=\displaystyle\frac{2x^m}{(2m+3)}(ax+b)^{3/2}&\\-\displaystyle\frac{2mb}{(2m+3)a}\displaystyle\int{x^{m-1}\displaystyle\sqrt{ax+b}dx} \end{align*}$

14.: $\displaystyle\int{\displaystyle\frac{\displaystyle\sqrt{ax+b}}{x^m}}=-\displaystyle\frac{\displaystyle\sqrt{ax+b}}{(m-1)x^{m-1}}+\displaystyle\frac{a}{2(m-1)}\displaystyle\int{\displaystyle\frac{dx}{x^{m-1}\displaystyle\sqrt{ax+b}}}$

15.: $\displaystyle\int{(ax+b)^{m/2}}=\displaystyle\frac{2(ax+b)^{(m+2)/2}}{a(m+2)}$

16.: $\displaystyle\int{x(ax+b)^{m/2}}=\displaystyle\frac{2(ax+b)^{(m+4)/2}}{a^2(m+4)}-\displaystyle\frac{2b(ax+b)^{(m+2)/2}}{a^2(m+2)}$

17.: $\begin{align*} \displaystyle\int{x^2(ax+b)^{m/2}}=\displaystyle\frac{2(ax+b)^{(m+6)/2}}{a^3(m+6)}-\displaystyle\frac{4b(ax+b)^{(m+4)/2}}{a^3(m+4)}&\\+\displaystyle\frac{2b^2(ax+b)^{(m+2)/2}}{a^3(m+2)} \end{align*}$

18.: $\displaystyle\int{\displaystyle\frac{(ax+b)^{m/2}}{x}dx}=\displaystyle\frac{2(ax+b)^{m/2}}{m}+b\displaystyle\int{\displaystyle\frac{(ax+b)^{(m-2)/2}}{x}dx}$

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