In order to determine the integral, we try a trig substitution. Since we have a 1+4x2
 term, we should use x=
(1/2) tan(θ)
.

This means that dx=
(1/2) sec(θ)^2
dθ
.

We have two possible options. We could determine the antiderivative in terms of θ
 and then convert back to x
 and then evaluate using our given bounds. Another option is that we could transform the x
 bounds on our original integral to θ
 bounds.

Using our substitution, we can convert our bounds.

When x=0
, θ=
0
.

When x=12
, θ=
π/4
.

Now we can express our original definite integral as a trigonometric integral in terms of the variable θ
.

∫1201+4x2−−−−−−√dx=∫
π/4
0
dθ

To solve the integral using the trigonometric substitution \( x = \frac{1}{2} \tan(\theta) \), we'll follow these steps:

### 1. Substitution setup:
- Given: \( 1 + 4x^2 \)
- Using the substitution \( x = \frac{1}{2} \tan(\theta) \), then:
  \[
  1 + 4x^2 = 1 + 4\left(\frac{1}{2} \tan(\theta)\right)^2 = 1 + \tan^2(\theta) = \sec^2(\theta)
  \]

- The derivative of \( x = \frac{1}{2} \tan(\theta) \) is:
  \[
  dx = \frac{1}{2} \sec^2(\theta) d\theta
  \]

### 2. Bounds conversion:
You’ve already correctly converted the bounds:
- When \( x = 0 \), \( \theta = 0 \)
- When \( x = 1/2 \), \( \theta = \frac{\pi}{4} \)

### 3. Rewrite the integral:
The original integral is:
\[
\int_0^{1/2} \sqrt{1 + 4x^2} \, dx
\]
Substitute \( x = \frac{1}{2} \tan(\theta) \), \( dx = \frac{1}{2} \sec^2(\theta) d\theta \), and \( 1 + 4x^2 = \sec^2(\theta) \). The integral becomes:
\[
\int_0^{\frac{\pi}{4}} \sqrt{\sec^2(\theta)} \cdot \frac{1}{2} \sec^2(\theta) \, d\theta
\]
Since \( \sqrt{\sec^2(\theta)} = \sec(\theta) \), the integral simplifies to:
\[
\frac{1}{2} \int_0^{\frac{\pi}{4}} \sec^3(\theta) \, d\theta
\]

### 4. Solving \( \int \sec^3(\theta) \, d\theta \):
To solve this integral, we use a standard technique for odd powers of secant. The integral of \( \sec^3(\theta) \) can be solved using integration by parts, or by using a known formula:
\[
\int \sec^3(\theta) \, d\theta = \frac{1}{2} \sec(\theta) \tan(\theta) + \frac{1}{2} \ln|\sec(\theta) + \tan(\theta)| + C
\]

Applying this to the limits from \( 0 \) to \( \frac{\pi}{4} \):
- At \( \theta = \frac{\pi}{4} \), \( \sec(\theta) = \sqrt{2} \) and \( \tan(\theta) = 1 \).
- At \( \theta = 0 \), \( \sec(0) = 1 \) and \( \tan(0) = 0 \).

Thus, the definite integral evaluates to:
\[
\frac{1}{2} \left[ \left( \frac{1}{2} \sec(\theta) \tan(\theta) + \frac{1}{2} \ln|\sec(\theta) + \tan(\theta)| \right) \right]_0^{\frac{\pi}{4}}
\]

### 5. Evaluation:
At \( \theta = \frac{\pi}{4} \):
- \( \sec(\frac{\pi}{4}) = \sqrt{2} \), \( \tan(\frac{\pi}{4}) = 1 \)
\[
\frac{1}{2} \left( \frac{1}{2} \sqrt{2} \cdot 1 + \frac{1}{2} \ln(\sqrt{2} + 1) \right)
\]

At \( \theta = 0 \):
- \( \sec(0) = 1 \), \( \tan(0) = 0 \)
\[
\frac{1}{2} \left( \frac{1}{2} \cdot 1 \cdot 0 + \frac{1}{2} \ln(1 + 0) \right) = 0
\]

Therefore, the result of the integral is:
\[
\frac{1}{2} \left( \frac{\sqrt{2}}{2} + \frac{1}{2} \ln(\sqrt{2} + 1) \right)
\]

Would you like to simplify this further or need any additional explanation?

### Follow-up questions:
1. How do you simplify trigonometric integrals with higher powers of secant or tangent?
2. What is the technique used to integrate \( \sec^3(\theta) \)?
3. Can trigonometric substitution be applied to other forms like \( \sqrt{a^2 - x^2} \)?
4. How do you handle bounds when switching between variables in definite integrals?
5. What are some common integrals that involve \( \sec(\theta) \) and \( \tan(\theta) \)?

### Tip:
When solving definite integrals with trigonometric substitution, always remember to carefully convert the bounds for consistency.

In order to determine the integral, we try a trig substitution. Since we have a 1+4x^2 term, we should use x = (1/2) tan(θ). This means that dx = (1/2) sec(θ)^2 dθ. We have two possible options. We could determine the antiderivative in terms of θ and then convert back to x and then evaluate using our given bounds. Another option is that we could transform the x bounds on our original integral to θ bounds. Using our substitution, we can convert our bounds. When x = 0, θ = 0. When x = 1/2, θ = π/4. Now we can express our original definite integral as a trigonometric integral in terms of the variable θ. ∫(0 to 1/2) sqrt(1 + 4x^2) dx = ∫(0 to π/4) dθ

This guide explains how to solve the definite integral ∫(0 to 1/2) sqrt(1 + 4x^2) dx using trigonometric substitution. Key steps include converting the integral into a trigonometric form and solving ∫ sec^3(θ) dθ. This solution covers advanced calculus topics and is suitable for undergraduate students.

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