https://file.aiquickdraw.com/mathbot/file/1726940338815az15f9ap.jpg

The integral you provided is:

\[
\int_{1/12}^{1/4} \frac{2 \, dt}{\sqrt{t} + 4t \sqrt{t}}
\]

### Step-by-Step Solution:

1. **Simplify the Denominator:**

   The denominator is \(\sqrt{t} + 4t\sqrt{t}\). Factor out \(\sqrt{t}\) from both terms:

   \[
   \sqrt{t} + 4t\sqrt{t} = \sqrt{t}(1 + 4t)
   \]

2. **Rewrite the Integral:**

   The integral now becomes:

   \[
   \int_{1/12}^{1/4} \frac{2 \, dt}{\sqrt{t}(1 + 4t)}
   \]

3. **Substitute** \(u = \sqrt{t}\), so that \(t = u^2\) and \(dt = 2u \, du\). The limits also change:
   - When \(t = \frac{1}{12}\), \(u = \sqrt{\frac{1}{12}} = \frac{1}{2\sqrt{3}}\)
   - When \(t = \frac{1}{4}\), \(u = \frac{1}{2}\)

   The integral becomes:

   \[
   \int_{\frac{1}{2\sqrt{3}}}^{\frac{1}{2}} \frac{2 \cdot 2u \, du}{u(1 + 4u^2)} = 4 \int_{\frac{1}{2\sqrt{3}}}^{\frac{1}{2}} \frac{du}{1 + 4u^2}
   \]

4. **Recognize the Standard Form:**

   The integral \(\int \frac{du}{1 + a^2u^2}\) is a standard arctangent formula:

   \[
   \int \frac{du}{1 + a^2u^2} = \frac{1}{a} \arctan(au)
   \]

   In this case, \(a = 2\), so the integral becomes:

   \[
   \frac{1}{2} \arctan(2u)
   \]

5. **Evaluate the Definite Integral:**

   Now apply the limits:

   \[
   4 \left[ \frac{1}{2} \arctan(2u) \right]_{\frac{1}{2\sqrt{3}}}^{\frac{1}{2}} = 2 \left[ \arctan(1) - \arctan\left(\frac{1}{\sqrt{3}}\right) \right]
   \]

   - \(\arctan(1) = \frac{\pi}{4}\)
   - \(\arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}\)

   So the result is:

   \[
   2 \left( \frac{\pi}{4} - \frac{\pi}{6} \right) = 2 \cdot \frac{\pi}{12} = \frac{\pi}{6}
   \]

### Final Answer:

\[
\boxed{\frac{\pi}{6}}
\]

Let me know if you'd like more details or have any further questions!

---

Here are five related questions to expand your understanding:

1. How can the arctangent integral formula be derived?
2. What is the geometric interpretation of arctan values in integrals?
3. Can this technique of substitution be applied to other radical expressions?
4. How do the limits of integration affect the final value in definite integrals?
5. What are other examples of integrals with trigonometric solutions?

**Tip:** When simplifying integrals, always look for possible factorizations or substitutions to make the process smoother.

Evaluate the integral from 1/12 to 1/4 of (2 dt) / (sqrt(t) + 4t sqrt(t)).

Solve the definite integral from 1/12 to 1/4 of (2 dt) / (sqrt(t) + 4t sqrt(t)) using substitution and the arctangent formula. Learn how to simplify the integral and apply standard trigonometric techniques in this step-by-step guide.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation