To solve the integral

$\int 2x(x^2 + 1)^2 \, dx$

we can use substitution to simplify the expression.

Step 1: Substitution

Let: $u = x^2 + 1$ Thus, the derivative of $u$ with respect to $x$ is: $\frac{du}{dx} = 2x \quad \Rightarrow \quad du = 2x \, dx$

Step 2: Rewriting the Integral

Substitute $u$ and $du$ into the integral: $\int 2x(x^2 + 1)^2 \, dx = \int u^2 \, du$

Step 3: Integrating

Now, integrate $u^2$: $\int u^2 \, du = \frac{u^3}{3} + C$

Step 4: Substituting Back

Substitute $u = x^2 + 1$ back into the result: $\frac{(x^2 + 1)^3}{3} + C$

Thus, the result of the integral is: $\frac{1}{3}(x^2 + 1)^3 + C$

Final Answer:

The correct answer is b: $\frac{1}{3}(x^2 + 1)^3 + C$

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Would you like further explanation or details on any part of the solution?

Here are some related questions for you to explore:

1. How would the result change if we had a different exponent on $(x^2 + 1)$?
2. What other integration techniques can be used for similar integrals?
3. Can you find the derivative of the result to verify the correctness of the integral?
4. How would the substitution method change if we had a term like $x^3$ instead of $2x$?
5. What are some other common types of integrals involving polynomials and rational functions?

Tip: When solving integrals involving polynomials raised to powers, substitution is often a very useful method for simplifying the expression before integration.