Matematica

Con $\small ax+b$

1.: $\displaystyle\int{\displaystyle\frac{dx}{ax+b}}=\displaystyle\frac{1}{a}\displaystyle\ln(ax+b)$

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

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

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

5.: $\displaystyle\int{\displaystyle\frac{dx}{x(ax+b)}}=\displaystyle\frac{1}{b}\displaystyle\ln(\displaystyle\frac{x}{ax+b})$

6.: $\displaystyle\int{\displaystyle\frac{dx}{x^2(ax+b)}}=-\displaystyle\frac{1}{bx}+\displaystyle\frac{a}{b^2}\ln\left(\displaystyle\frac{ax+b}{x}\right)$

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

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

9.: $\displaystyle\int{\displaystyle\frac{xdx}{(ax+b)^2}}=\displaystyle\frac{b}{a^2(ax+b)}+\displaystyle\frac{1}{a^2}\ln\left(ax+b\right)$

10.: $\displaystyle\int{\displaystyle\frac{x^2dx}{(ax+b)^2}}=\displaystyle\frac{ax+b}{a^3}-\displaystyle\frac{b^2}{a^3(ax+b)}-\displaystyle\frac{2b}{a^3}\ln\left(ax+b\right)$

11.: $\begin{align*} \displaystyle\int{\displaystyle\frac{x^3dx}{(ax+b)^2}}=\displaystyle\frac{(ax+b)^2}{2a^4}-\displaystyle\frac{3b(ax+b)}{a^4}+\displaystyle\frac{b^3}{a^4(ax+b)} &\\+\displaystyle\frac{3b^2}{a^4}\ln\left(ax+b\right) \end{align*}$

12.: $\displaystyle\int{\displaystyle\frac{dx}{x(ax+b)^2}}=\displaystyle\frac{1}{b(ax+b)}+\displaystyle\frac{1}{b^2}\displaystyle\ln\left(\frac{x}{ax+b}\right)$

13.: $\displaystyle\int{\displaystyle\frac{dx}{x^2(ax+b)^2}}=\displaystyle\frac{-a}{b^2(ax+b)}-\displaystyle\frac{1}{b^2x}+\displaystyle\frac{2a}{b^3}\displaystyle\ln\left(\displaystyle\frac{ax+b}{x}\right)$

14.: $\begin{align*} \displaystyle\int{\displaystyle\frac{dx}{x^3(ax+b)^2}}=-\displaystyle\frac{(ax+b)^2}{2b^4x^2}+\displaystyle\frac{3a(ax+b)}{b^4x}-\displaystyle\frac{a^3x}{b^4(ax+b)} &\\-\displaystyle\frac{3a^2}{b^4}\displaystyle\ln\left(\displaystyle\frac{ax+b}{x}\right) \end{align*} $

15.: $\displaystyle\int{\displaystyle\frac{dx}{(ax+b)^3}}=\displaystyle\frac{-1}{2a(ax+b)^2}$

16.: $\displaystyle\int{\displaystyle\frac{xdx}{(ax+b)^3}}=\displaystyle\frac{-1}{a^2(ax+b)}+\displaystyle\frac{b}{2a^2(ax+b)^2}$

17.: $\displaystyle\int{\displaystyle\frac{x^2dx}{(ax+b)^3}}=\displaystyle\frac{2b}{a^3(ax+b)}-\displaystyle\frac{b^2}{2a^3(ax+b)^2}+\displaystyle\frac{1}{a^3}\displaystyle\ln(ax+b)$

18.: $\displaystyle\int{\displaystyle\frac{x^3dx}{(ax+b)^3}}=\displaystyle\frac{x}{a^3}-\displaystyle\frac{3b^2}{a^4(ax+b)}+\displaystyle\frac{b^3}{2a^4(ax+b)^2}-\displaystyle\frac{3b}{a^4}\displaystyle\ln(ax+b)$

19.: $\displaystyle\int{\displaystyle\frac{dx}{x(ax+b)^3}}=\displaystyle\frac{a^2x^2}{2b^3(ax+b)^2}-\displaystyle\frac{2ax}{b^3(ax+b)}-\displaystyle\frac{1}{b^3}\displaystyle\ln\left(\displaystyle\frac{ax+b}{x}\right)$

20.: $\begin{align*} \displaystyle\int{\displaystyle\frac{dx}{x^2(ax+b)^3}}=\displaystyle\frac{-a}{2b^2(ax+b)^2}-\displaystyle\frac{2a}{b^3(ax+b)}-\displaystyle\frac{1}{b^3x}&\\+\displaystyle\frac{3a}{b^4}\displaystyle\ln\left(\displaystyle\frac{ax+b}{x}\right) \end{align*}$

21.: $\begin{align*} \displaystyle\int{\displaystyle\frac{dx}{x^3(ax+b)^3}}=\displaystyle\frac{a^4x^2}{2b^5(ax+b)^2}-\displaystyle\frac{4a^3x}{b^5(ax+b)}-\displaystyle\frac{(ax+b)^2}{2b^5x^2} &\\ -\displaystyle\frac{6a^2}{b^5}\displaystyle\ln\left(\displaystyle\frac{ax+b}{x}\right) \end{align*} $

22.: $\displaystyle\int{(ax+b)^ndx}=\displaystyle\frac{(ax+b)^{n+1}}{(n+1)a},\quad n\neq-1$

23.: $\displaystyle\int{x(ax+b)^ndx}=\displaystyle\frac{(ax+b)^{n+2}}{(n+2)a^2}-\displaystyle\frac{b(ax+b)^{n+1}}{(n+1)a^2},\quad n\neq-1,-2$

24.: $\displaystyle\int{x^2(ax+b)^ndx}=\displaystyle\frac{(ax+b)^{n+3}}{(n+3)a^3}-\displaystyle\frac{2b(ax+b)^{n+2}}{(n+2)a^3}+\displaystyle\frac{b^2(ax+b)^{n+1}}{(n+1)a^3},$

25.: $\int x^{m}(ax+b)^{n}\,dx= \left\{\begin{matrix} \frac{x^{m+1}\left ( ax+b \right )^{n}}{m+n+1}+\frac{nb}{m+n+1}\int x^m\left ( ax+b \right )^{n-1}dx \\ \frac{x^m\left ( ax+b \right )^{n+1}}{\left ( m+n+1 \right )a}-\frac{mb}{\left ( m+n+1 \right )a}\int x^{m-1}\left ( ax+b \right )^ndx \\ \frac{-x^{m+1}\left ( ax+b \right )^{n+1}}{\left ( n+1 \right )^b}+\frac{m+n+2}{\left ( n+1 \right )b}\int x^m\left ( ax+b \right )^{n+1} dx \end{matrix}\right.$