https://r2.erweima.ai/mathbot/image/1726952372148dcs95gnx.jpg

The solution provided in the image aims to evaluate the integral:

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

Here's a breakdown of the steps:

### Step 1: Substitution
They define:

\[
x^2 = \tan \theta \quad \Rightarrow \quad \theta = \tan^{-1} x^2 \quad \Rightarrow \quad x \, dx = \frac{1}{2} \sec^2 \theta \, d\theta
\]

This is a correct substitution to help simplify the integral.

### Step 2: Using the Substitution
Substitute this into the original integral:

\[
\int \frac{x \, dx}{1 + x^4} = \int \frac{\frac{1}{2} \sec^2 \theta \, d\theta}{1 + \tan^2 \theta}
\]

Since \( 1 + \tan^2 \theta = \sec^2 \theta \), the integral simplifies to:

\[
\int \frac{\frac{1}{2} \sec^2 \theta \, d\theta}{\sec^2 \theta} = \frac{1}{2} \int d\theta = \frac{1}{2} \theta + C
\]

### Step 3: Back Substitution
From the earlier substitution \( \theta = \tan^{-1} x^2 \), we substitute back:

\[
\frac{1}{2} \theta + C = \frac{1}{2} \tan^{-1} x^2 + C
\]

Thus, the integral evaluates to:

\[
\int \frac{x}{1+x^4} \, dx = \frac{1}{2} \tan^{-1} x^2 + C
\]

### Conclusion:
The solution in the image correctly evaluates the integral, so the statement is **verdadero (true)**.

Would you like more details or have any questions? Here are five related questions for further practice:

1. What is the integral of \( \int \frac{1}{1+x^4} \, dx \)?
2. How does trigonometric substitution help in solving integrals involving powers of \( x \)?
3. Can you solve \( \int \frac{x^2}{1+x^4} \, dx \)?
4. What is the geometric interpretation of \( \tan^{-1} x^2 \) in the context of integrals?
5. How do different substitution methods compare when evaluating complex integrals?

**Tip:** When using trigonometric substitution, always check if the identity \(1 + \tan^2 \theta = \sec^2 \theta\) simplifies your expression directly.

This solution demonstrates the evaluation of the integral ∫(x/(1+x^4)) dx using trigonometric substitution. The problem involves substitution with x^2 = tan(θ) and utilizes the identity 1 + tan^2(θ) = sec^2(θ) to simplify the integral. Suitable for undergraduate calculus students.

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