# How do you integrate (4x^2 + 6x^(3/2) + 13x + 5)/sqrtx?

Apr 16, 2016

$\frac{8}{5} {x}^{\frac{5}{2}} + 3 {x}^{2} + \frac{26}{3} {x}^{\frac{3}{2}} + 10 {x}^{\frac{1}{2}} + C$

#### Explanation:

Rearrange the algebra to make it all one continuous line, which is far easier to integrate than having a fraction, knowing that $\sqrt{x} = {x}^{\frac{1}{2}}$

$\frac{4 {x}^{2} + 6 {x}^{\frac{3}{2}} + 13 x + 5}{x} ^ \left(\frac{1}{2}\right) = 4 {x}^{2 - \frac{1}{2}} + 6 {x}^{\frac{3}{2} - \frac{1}{2}} + 13 {x}^{1 - \frac{1}{2}} + 5 {x}^{0 - \frac{1}{2}}$
$= 4 {x}^{\frac{3}{2}} + 6 x + 13 {x}^{\frac{1}{2}} + 5 {x}^{- \frac{1}{2}}$

Now this is far easier to integrate. Increase each power by one and divide by the new power, doing each phrase individually and putting them back together at the end. At this point we can leave out the constant of integration, but make sure to replace it at the end.

$\int 4 {x}^{\frac{3}{2}} \mathrm{dx} = \frac{4 {x}^{\frac{5}{2}}}{\frac{5}{2}} = \frac{8}{5} {x}^{\frac{5}{2}}$
$\int 6 x \mathrm{dx} = 3 {x}^{2}$
$\int 13 {x}^{\frac{1}{2}} \mathrm{dx} = \frac{26}{3} {x}^{\frac{3}{2}}$
$\int 5 {x}^{- \frac{1}{2}} \mathrm{dx} = 10 {x}^{\frac{1}{2}}$

Now put these back together, and add in the constant of integration,

$\frac{8}{5} {x}^{\frac{5}{2}} + 3 {x}^{2} + \frac{26}{3} {x}^{\frac{3}{2}} + 10 {x}^{\frac{1}{2}} + C$