Evaluate the integral of (3√(1 + 4√x) / √x) dx.

Steps to Solve

1. Rewrite the Integral We express the integral in a simpler form. The given integral is

$$ \int \frac{3\sqrt{(1 + 4\sqrt{x})}}{\sqrt{x}} , dx $$

This can be rewritten as:

$$ 3 \int \frac{\sqrt{1 + 4\sqrt{x}}}{\sqrt{x}} , dx $$

2. Substitute for Simplicity Let’s substitute $u = \sqrt{x}$, which means that $x = u^2$ and $dx = 2u , du$. Then, the integral transforms to:

$$ 3 \int \frac{\sqrt{1 + 4u}}{u} \cdot (2u) , du $$

This simplifies to:

$$ 6 \int (1 + 4u)^{1/2} , du $$

3. Expand the Integral Next, we can break this integral down using the binomial expansion:

$$ 6 \int (1 + 4u)^{1/2} , du = 6 \int (1 + 4u)^{1/2} , du $$

Using the substitution method: Let $v = 1 + 4u$, then $dv = 4,du \Rightarrow du = \frac{1}{4}dv$.

4. Integral Transformation Substituting gives us:

$$ 6 \cdot \frac{1}{4} \int v^{1/2} , dv = \frac{3}{2} \int v^{1/2} , dv $$

5. Integrate Now, we can integrate $v^{1/2}$:

$$ \frac{3}{2} \cdot \frac{2}{3} v^{3/2} + C = v^{3/2} + C $$

Substituting back $v = 1 + 4u$ gives:

$$ (1 + 4u)^{3/2} + C $$

6. Re-substitute for x Finally, we replace $u$ back with $\sqrt{x}$:

$$ (1 + 4\sqrt{x})^{3/2} + C $$

Thus, the final answer for the integral is:

$$ \int \frac{3\sqrt{(1 + 4\sqrt{x})}}{\sqrt{x}} , dx = (1 + 4\sqrt{x})^{3/2} + C $$