To solve the integral of $8x \sqrt{4 + 2x^2} \, dx$ using $u$-substitution, we follow these steps:

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Step 1: Choose a substitution

Let: $u = 4 + 2x^2$ This is chosen because the square root $\sqrt{4 + 2x^2}$ involves a composite function, and $2x^2$ is its derivative component.

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Step 2: Differentiate $u$ to express $dx$

Differentiating $u$ with respect to $x$, we get: $\frac{du}{dx} = 4x \quad \implies \quad du = 4x \, dx$ Notice that $8x \, dx$ in the integral can now be replaced using $du$.

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Step 3: Rewrite the integral in terms of $u$

Substituting $u = 4 + 2x^2$ and $du = 4x \, dx$, the integral becomes: $\int 8x \sqrt{4 + 2x^2} \, dx = \int 2 \sqrt{u} \, du$

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Step 4: Simplify and evaluate the integral

The simplified integral is: $\int 2 \sqrt{u} \, du = 2 \int u^{1/2} \, du$ Use the power rule for integration: $\int u^{n} \, du = \frac{u^{n+1}}{n+1} \quad (\text{for } n \neq -1)$ Here, $n = 1/2$, so: $2 \int u^{1/2} \, du = 2 \cdot \frac{u^{3/2}}{3/2} = \frac{4}{3} u^{3/2}$

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Step 5: Substitute back $u = 4 + 2x^2$

Returning to the original variable $x$: $\frac{4}{3} u^{3/2} = \frac{4}{3} (4 + 2x^2)^{3/2}$

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Final Answer:

$\int 8x \sqrt{4 + 2x^2} \, dx = \frac{4}{3} (4 + 2x^2)^{3/2} + C$ where $C$ is the constant of integration.

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Key Points:

1. The substitution $u = 4 + 2x^2$ simplifies the integral by targeting the inner composite function.
2. The derivative of $u$, $du = 4x \, dx$, helps to replace the $x$-related terms.
3. The power rule for integration is used to handle the new integral in $u$.

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Do you want additional details or a visual breakdown?

Here are 5 related questions:

1. How do you apply $u$-substitution when the function involves trigonometric components?
2. What other methods could you use if $u$-substitution is not obvious?
3. How does the constant of integration affect definite integrals?
4. What is the geometric interpretation of this integral?
5. Can we generalize this substitution technique for $a + bx^n$ under a square root?

Tip: Always verify $du$ matches exactly with terms in your integral to avoid errors in substitution.