The integral you've provided is:

$\int \frac{8x + 4}{\sqrt{4x - 3}} \, dx$

To solve this, let's break it down step by step.

Step 1: Simplify the Integrand

Notice that the numerator can be split into two parts:

$\int \frac{8x + 4}{\sqrt{4x - 3}} \, dx = \int \frac{8x}{\sqrt{4x - 3}} \, dx + \int \frac{4}{\sqrt{4x - 3}} \, dx$

Step 2: Solve Each Integral Separately

Integral 1: $\int \frac{8x}{\sqrt{4x - 3}} \, dx$

Let $u = 4x - 3$, then $du = 4 \, dx$, and $dx = \frac{du}{4}$. Also, $x = \frac{u + 3}{4}$.

Substituting these into the integral:

$\int \frac{8x}{\sqrt{4x - 3}} \, dx = \int \frac{8 \cdot \frac{u + 3}{4}}{\sqrt{u}} \cdot \frac{du}{4} = 2 \int \frac{u + 3}{\sqrt{u}} \, du$

This can be further simplified:

$= 2 \int \left( \frac{u}{\sqrt{u}} + \frac{3}{\sqrt{u}} \right) du = 2 \int \left( u^{1/2} + 3u^{-1/2} \right) du$

Now, integrate term by term:

$= 2 \left( \frac{2}{3} u^{3/2} + 6u^{1/2} \right) = \frac{4}{3} u^{3/2} + 12u^{1/2}$

Substitute back $u = 4x - 3$:

$= \frac{4}{3} (4x - 3)^{3/2} + 12(4x - 3)^{1/2}$

Integral 2: $\int \frac{4}{\sqrt{4x - 3}} \, dx$

Using the same substitution $u = 4x - 3$ and $dx = \frac{du}{4}$:

$\int \frac{4}{\sqrt{4x - 3}} \, dx = \int \frac{4}{\sqrt{u}} \cdot \frac{du}{4} = \int u^{-1/2} \, du$

The integral of $u^{-1/2}$ is:

$2u^{1/2} = 2\sqrt{4x - 3}$

Step 3: Combine the Results

Adding the results of both integrals:

$\frac{4}{3} (4x - 3)^{3/2} + 12(4x - 3)^{1/2} + 2\sqrt{4x - 3} + C$

Simplifying:

$\int \frac{8x + 4}{\sqrt{4x - 3}} \, dx = \frac{4}{3} (4x - 3)^{3/2} + 14\sqrt{4x - 3} + C$

Where $C$ is the constant of integration.

Would you like more details on any step or have any questions?

Related Questions:

1. How would the integration change if the numerator was $8x + 2$ instead of $8x + 4$?
2. What technique could be used if the integrand was $\frac{8x + 4}{\sqrt{4x^2 - 3}}$ instead?
3. How does the choice of substitution simplify the process in solving integrals?
4. What is the significance of the constant of integration $C$ in indefinite integrals?
5. How would this integral be approached using a numerical method like Simpson's Rule?

Tip:

When faced with complex integrals, look for algebraic simplifications or substitutions that can reduce the problem to a more standard form.