# What is the antiderivative of (2x)/(sqrt(5 + 4x))?

Mar 31, 2018

$\frac{\sqrt{5 + 4 x}}{6} \left(2 x - 5\right) + c$

#### Explanation:

To find the antiderivertive of $\frac{2 x}{\sqrt{5 + 4 x}}$ we calculate the integral

$\int \left(\frac{2 x}{\sqrt{5 + 4 x}}\right) \mathrm{dx}$

substitute $u = 5 + 4 x$ and $\mathrm{du} = 4 \mathrm{dx}$
$\int \left(\frac{u - 5}{8 \sqrt{u}}\right) \mathrm{du} = \frac{1}{8} \left(\int \left(\frac{u}{\sqrt{u}}\right) \mathrm{du} - \int \left(\frac{5}{\sqrt{u}}\right) \mathrm{du}\right)$

$\int \left(\frac{5}{\sqrt{u}}\right) \mathrm{du} = 10 \sqrt{u} = 10 \sqrt{5 + 4 x}$
$\int \left(\frac{u}{\sqrt{u}}\right) \mathrm{du} = \int \left(\sqrt{u}\right) \mathrm{du} = \frac{2}{3} \sqrt{{u}^{3}} = \frac{2}{3} \sqrt{{\left(5 + 4 x\right)}^{3}}$
All in all:

$\frac{\sqrt{5 + 4 x}}{6} \left(2 x - 5\right) + c$

Mar 31, 2018

$\int \frac{2 x}{\sqrt{5 + 4 x}} \mathrm{dx} = \frac{1}{6} \left(2 x - 5\right) \sqrt{5 + 4 x} + C$

#### Explanation:

$\int \frac{2 x}{\sqrt{5 + 4 x}} \mathrm{dx}$

Let's first substitute

$u = 5 + 4 x$

$\Rightarrow \mathrm{du} = 4 \mathrm{dx}$

$\frac{1}{4} \mathrm{du} = \mathrm{dx}$

$\Rightarrow u - 5 = 4 x$

$\Rightarrow \frac{1}{2} \left(u - 5\right) = 2 x$

Now the integral becomes:

$\int \frac{\frac{1}{2} \left(u - 5\right)}{\sqrt{u}} \frac{1}{4} \mathrm{du}$

$\Rightarrow \frac{1}{8} \int \frac{u - 5}{\sqrt{u}} \mathrm{du}$

Let's split this into two simpler integrals:

$\Rightarrow \frac{1}{8} \int \left(\frac{u}{\sqrt{u}} - \frac{5}{\sqrt{u}}\right) \mathrm{du}$

$\Rightarrow \frac{1}{8} \left[\int \sqrt{u} \mathrm{du} - 5 \int \frac{1}{\sqrt{u}} \mathrm{du}\right]$

... and let's express the roots as exponents to make integrating more intuitive.

$\Rightarrow \frac{1}{8} \left[\int {u}^{\frac{1}{2}} \mathrm{du} - 5 \int {u}^{- \frac{1}{2}} \mathrm{du}\right]$

Integrating, we get:

$\Rightarrow \frac{1}{8} \left[{u}^{\frac{3}{2}} / \left(\frac{3}{2}\right) - 5 {u}^{\frac{1}{2}} / \left(\frac{1}{2}\right)\right] + C$

$\Rightarrow \frac{1}{8} \left[\frac{2}{3} {u}^{\frac{3}{2}} - 10 {u}^{\frac{1}{2}}\right] + C$

$\Rightarrow \frac{2}{24} {u}^{\frac{3}{2}} - \frac{10}{8} {u}^{\frac{1}{2}} + C$

$\Rightarrow \frac{1}{12} {u}^{\frac{3}{2}} - \frac{5}{4} {u}^{\frac{1}{2}} + C$

$\Rightarrow \frac{1}{4} {u}^{\frac{1}{2}} \left(\frac{1}{3} u - 5\right) + C$

Now let's express the answer in terms of $x$, and perform some algebra to simplify the expression.

$\Rightarrow \frac{1}{4} \sqrt{5 + 4 x} \left(\frac{1}{3} \left(5 + 4 x\right) - 5\right) + C$

$\Rightarrow \frac{1}{4} \sqrt{5 + 4 x} \left(\frac{1}{3} \left(5 + 4 x - 15\right)\right) + C$

$\Rightarrow \frac{1}{4} \sqrt{5 + 4 x} \left(\frac{1}{3} \left(4 x - 10\right)\right) + C$

$\Rightarrow \frac{1}{12} \left(4 x - 10\right) \sqrt{5 + 4 x} + C$

$\Rightarrow \frac{1}{12} \cdot 2 \left(2 x - 5\right) \sqrt{5 + 4 x} + C$

$\Rightarrow \frac{1}{6} \left(2 x - 5\right) \sqrt{5 + 4 x} + C$