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The given integral is:

\[
\int_0^1 \sin^{-1} \left( \frac{2x}{1 + x^2} \right) dx
\]

We are tasked with evaluating this using an appropriate substitution method. A suitable substitution would involve simplifying the term inside the inverse sine function.

### Step-by-Step Solution

Let's try using the substitution:
\[
x = \tan(\theta), \quad dx = \sec^2(\theta) d\theta
\]
This substitution helps simplify expressions involving \(1 + x^2\).

#### Step 1: Substituting into the integral

Using the substitution \(x = \tan(\theta)\):
\[
1 + x^2 = 1 + \tan^2(\theta) = \sec^2(\theta)
\]
Thus, the expression \(\frac{2x}{1 + x^2}\) becomes:
\[
\frac{2\tan(\theta)}{\sec^2(\theta)} = 2\sin(\theta)\cos(\theta)
\]

This simplifies the original expression for the inverse sine:
\[
\sin^{-1}\left(2\sin(\theta)\cos(\theta)\right) = \sin^{-1}(\sin(2\theta)) = 2\theta
\]

#### Step 2: Transform the limits of integration

When \(x = 0\), \(\theta = 0\).
When \(x = 1\), \(\theta = \frac{\pi}{4}\).

The new limits of integration are from \(0\) to \(\frac{\pi}{4}\), and the integral becomes:
\[
\int_0^{\frac{\pi}{4}} 2\theta \cdot \sec^2(\theta) d\theta
\]

#### Step 3: Solve the transformed integral

We now have to solve:
\[
2 \int_0^{\frac{\pi}{4}} \theta \sec^2(\theta) d\theta
\]

This can be solved using integration by parts. Let:
\[
u = \theta, \quad dv = \sec^2(\theta) d\theta
\]
Then:
\[
du = d\theta, \quad v = \tan(\theta)
\]

Now apply the integration by parts formula:
\[
\int u dv = uv - \int v du
\]
Substituting:
\[
2\left[\theta \tan(\theta) \Big|_0^{\frac{\pi}{4}} - \int_0^{\frac{\pi}{4}} \tan(\theta) d\theta\right]
\]

Evaluate each part:
1. \(\theta \tan(\theta) \Big|_0^{\frac{\pi}{4}} = \frac{\pi}{4}\)
2. \(\int_0^{\frac{\pi}{4}} \tan(\theta) d\theta = \ln(\sec(\theta)) \Big|_0^{\frac{\pi}{4}} = \ln(\sec(\frac{\pi}{4})) = \ln(\sqrt{2}) = \frac{1}{2} \ln(2)\)

Thus, the integral evaluates to:
\[
2\left[\frac{\pi}{4} - \frac{1}{2} \ln(2)\right]
= \frac{\pi}{2} - \ln(2)
\]

### Final Answer:
\[
\boxed{\frac{\pi}{2} - \ln(2)}
\]

Let me know if you would like more details or have any further questions! Here are five related questions for further practice:

1. How do you choose a substitution in integration problems?
2. What is the significance of inverse trigonometric functions in integrals?
3. How does integration by parts work, and when should it be used?
4. What is the integral of \(\sec^2(x)\)?
5. Can you evaluate an integral involving inverse sine without substitution?

**Tip:** In problems involving trigonometric identities and substitution, simplifying the argument of inverse trigonometric functions often helps.

Using any suitable integration technique, evaluate ∫ from 0 to 1 of sin^(-1)((2x)/(1+x^2)) dx, giving your answer in exact form.

Learn how to evaluate the integral ∫ from 0 to 1 of sin^(-1)((2x)/(1+x^2)) dx using substitution and integration by parts. This solution covers inverse trigonometric functions, substitution techniques, and integration by parts. Suitable for university-level calculus students.

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