https://file.aiquickdraw.com/mathbot/file/17254568957332dcjrnqa.jpg

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.

Evaluate the integral \( \int \frac{8x + 4}{\sqrt{4x - 3}} \, dx \)

This page provides a detailed solution for the integral \( \int \frac{8x + 4}{\sqrt{4x - 3}} \, dx \), involving algebraic simplification and the substitution method. The problem is suitable for undergraduate students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation