https://r2.erweima.ai/mathbot/image/1725827904284p5hxmxy5.jpg

The integral you've provided is:

\[
\int 4 \sqrt{4x + 1} \, dx
\]

Let's solve this step by step.

### Step 1: Substitution
We can use substitution to simplify the square root term. Let:
\[
u = 4x + 1
\]
Then, the derivative \( du \) will be:
\[
du = 4 \, dx \quad \text{or} \quad dx = \frac{du}{4}
\]

### Step 2: Rewrite the Integral
Substituting the values of \( u \) and \( dx \) into the integral, we get:
\[
\int 4 \sqrt{u} \cdot \frac{du}{4}
\]
The factors of 4 cancel out, so we are left with:
\[
\int \sqrt{u} \, du
\]

### Step 3: Solve the Integral
The integral of \( \sqrt{u} \) is:
\[
\int u^{1/2} \, du = \frac{2}{3} u^{3/2} + C
\]

### Step 4: Back-Substitute for \( u \)
Now, substitute back \( u = 4x + 1 \):
\[
\frac{2}{3} (4x + 1)^{3/2} + C
\]

### Final Answer:
Thus, the solution to the integral is:
\[
\frac{2}{3} (4x + 1)^{3/2} + C
\]

If you have any questions or need further clarification, feel free to ask!

---

**Next steps or related questions:**
1. What if the coefficient of \( x \) was different in the square root?
2. How would you handle this integral if the power of the expression inside the root were different?
3. How does substitution work for more complex integrals?
4. Can integrals always be solved by substitution, or are there cases where different techniques are required?
5. What is the significance of the constant of integration?

**Tip:** Always double-check your substitution to ensure all variables have been correctly transformed, especially when handling constants.

Learn how to solve the integral ∫ 4√(4x + 1) dx using the substitution method. This problem involves integral calculus and is suitable for undergraduate students. Step-by-step instructions and explanations are provided, including the application of the substitution method and power rule for integration.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation