INTEGRALS CONTAINING THE SQUARE ROOT OF ax+b

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

2.
$\displaystyle \int\displaystyle \frac{x\,dx}{\displaystyle \sqrt{ax+b}}=\displaystyle \frac{2(ax-2b)}{3a^{\displaystyle2}}\displaystyle \sqrt{ax+b}$

3.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{\displaystyle \sq...
...le2}-4abx+8b^{\displaystyle
2})}{15a^{\displaystyle3}}\displaystyle \sqrt{ax+b}$

4.
$\displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{ax+b}}=\left\{ \...
...tyle-1}\displaystyle \sqrt{\displaystyle \frac{ax+b}{-b}}
\end{array} \right. $

5.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle
2}\displaystyle \sqr...
...rac{a}{2b}\displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{ax+b}}$

6.
$\displaystyle \int\displaystyle \sqrt{ax+b}\,dx=\displaystyle \frac{2\displaystyle \sqrt{(ax+b)^{\displaystyle3}}}{3a}$

7.
$\displaystyle \int x\displaystyle \sqrt{ax+b}\,dx=\displaystyle \frac{2(3ax-2b)}{15a^{\displaystyle2}}\displaystyle \sqrt{(ax+b)^{\displaystyle3}}$

8.
$\displaystyle \int x^{\displaystyle2}\displaystyle \sqrt{ax+b}\,dx=\displaystyl...
...aystyle
2})}{105a^{\displaystyle3}}\displaystyle \sqrt{(a+bx)^{\displaystyle3}}$

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

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

11.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle m}}{\displaystyle \sqrt{...
...e \int\displaystyle \frac{x^{\displaystyle
m-1}}{\displaystyle \sqrt{ax+b}}\,dx$

12.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}\displaystyle \sqr...
...yle \int\displaystyle \frac{dx}{x^{\displaystyle m-1}\displaystyle \sqrt{ax+b}}$

13.
$\displaystyle \int x^{\displaystyle m}\displaystyle \sqrt{ax+b}\,dx=\displaysty...
...}{(2m+3)a}\displaystyle \int x^{\displaystyle m-1}\displaystyle \sqrt{ax+b}\,dx$

14.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{ax+b}}{x^{\displaysty...
...yle \int\displaystyle \frac{dx}{x^{\displaystyle m-1}\displaystyle \sqrt{ax+b}}$

15.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{ax+b}}{x^{\displaysty...
...e \int\displaystyle \frac{\displaystyle \sqrt{ax+b}}{x^{\displaystyle m-1}}\,dx$

16.
$\displaystyle \int(ax+b)^{\displaystyle m/2}\,dx=\displaystyle \frac{2(ax+b)^{\displaystyle(m+2)/2}}{a(m+2)}$

17.
$\displaystyle \int x(ax+b)^{\displaystyle m/2}\,dx=\displaystyle \frac{2(ax+b)^...
...}-\displaystyle \frac{2b(ax+b)^{\displaystyle(m+2)/2}}{a^{\displaystyle2}(m+2)}$

18.
$\begin{array}{lcl}
\displaystyle \int x^{\displaystyle2}(ax+b)^{\displaystyle m...
...splaystyle2}(ax+b)^{\displaystyle(m+2)/2}}{a^{\displaystyle3}(m+2)}
\end{array}$

19.
$\displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle m/2}}{x}\,dx=\displ...
...m}+b\displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle(m-2)/2}}{x}\,dx$

20.
$\displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle m/2}}{x^{\displayst...
...ma}{2b}\displaystyle \int\displaystyle \frac{(ax+b)^{\displaystyle m/2}}{x}\,dx$

21.
$\displaystyle \int\displaystyle \frac{dx}{x(ax+b)^{\displaystyle m/2}}=\display...
...{1}{b}\displaystyle \int\displaystyle \frac{dx}{x(ax+b)^{\displaystyle(m-2)/2}}$

[Tables]

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