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.

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