Integral of 1/(route 2x+5)+(route 2x-3)?

1/$\sqrt{2 x + 5}$+sqrt(2x-3

Mar 26, 2018

$I = \frac{1}{24} \left({\left(2 x + 5\right)}^{\frac{3}{2}} - {\left(2 x - 3\right)}^{\frac{3}{2}}\right) + C$

Explanation:

We want to solve

$I = \int \frac{1}{\sqrt{2 x + 5} + \sqrt{2 x - 3}} \mathrm{dx}$

Rationalize the denominator of the integrand

$I = \int \frac{1}{\sqrt{2 x + 5} + \sqrt{2 x - 3}} \cdot \frac{\sqrt{2 x + 5} - \sqrt{2 x - 3}}{\sqrt{2 x + 5} - \sqrt{2 x - 3}} \mathrm{dx}$

$\textcolor{w h i t e}{I} = \int \frac{\sqrt{2 x + 5} - \sqrt{2 x - 3}}{2 x + 5 - \left(2 x - 3\right)} \mathrm{dx}$

$\textcolor{w h i t e}{I} = \frac{1}{8} \int \sqrt{2 x + 5} - \sqrt{2 x - 3} \mathrm{dx}$

Make a substitution color(brown)(u=2x=>du=2dx

$I = \frac{1}{16} \int \sqrt{u + 5} - \sqrt{u - 3} \mathrm{dx}$

$\textcolor{w h i t e}{I} = \frac{1}{16} \left(\frac{2}{3} {\left(u + 5\right)}^{\frac{3}{2}} - \frac{2}{3} {\left(u - 3\right)}^{\frac{3}{2}}\right) + C$

$\textcolor{w h i t e}{I} = \frac{1}{24} \left({\left(u + 5\right)}^{\frac{3}{2}} - {\left(u - 3\right)}^{\frac{3}{2}}\right) + C$

Substitute back $u = 2 x$

$I = \frac{1}{24} \left({\left(2 x + 5\right)}^{\frac{3}{2}} - {\left(2 x - 3\right)}^{\frac{3}{2}}\right) + C$