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To solve the integral 

\[
\int_0^2 \frac{x \, dx}{\sqrt{1 + 2x^2}}
\]

we can approach it by using a trigonometric substitution or a substitution that simplifies the expression inside the square root.

### Step-by-Step Solution:

1. **Substitute**: Let's try the substitution \( u = 1 + 2x^2 \), which simplifies the square root term.
   - Then, \( du = 4x \, dx \) or \( \frac{du}{4} = x \, dx \).
   
2. **Rewrite the Integral in Terms of \( u \)**:
   - When \( x = 0 \): \( u = 1 \).
   - When \( x = 2 \): \( u = 1 + 2(2^2) = 1 + 8 = 9 \).
   - So the integral becomes:
     \[
     \int_1^9 \frac{1}{\sqrt{u}} \cdot \frac{du}{4} = \frac{1}{4} \int_1^9 u^{-\frac{1}{2}} \, du.
     \]

3. **Integrate**:
   - The antiderivative of \( u^{-\frac{1}{2}} \) is \( 2u^{\frac{1}{2}} \), so we have:
     \[
     \frac{1}{4} \cdot 2 u^{\frac{1}{2}} \Big|_1^9 = \frac{1}{2} \left( u^{\frac{1}{2}} \Big|_1^9 \right).
     \]
   - Substitute the limits:
     \[
     \frac{1}{2} \left( \sqrt{9} - \sqrt{1} \right) = \frac{1}{2} (3 - 1) = \frac{1}{2} \cdot 2 = 1.
     \]

### Final Answer:
The value of the integral is \( \boxed{1} \), which corresponds to option **d**.

Would you like further details on any step, or have any questions about this process?

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Here are five related questions to deepen your understanding:

1. How would the solution change if the integral limits were different?
2. Can a different substitution method, such as trigonometric substitution, also work here?
3. What are other examples where substitution simplifies an integral with square roots?
4. How would this integral look if the denominator had a different expression, like \(1 + x^2\)?
5. How does integration by parts apply to integrals with a similar form?

**Tip:** When dealing with square roots in integrals, try substitution to simplify the expression under the square root first.

Evaluate the integral \( \int_0^2 \frac{x \, dx}{\sqrt{1 + 2x^2}} \) and choose the correct answer from the options given.

Find the value of the integral \( \int_0^2 \frac{x \, dx}{\sqrt{1 + 2x^2}} \) using substitution techniques in calculus. A step-by-step solution is provided to determine the correct answer.

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