To solve the integral $\int 2x(x^2 + 1) \, dx$, we'll follow these steps:

Step 1: Expand the Integrand

First, distribute $2x$ across $(x^2 + 1)$:

$\int 2x(x^2 + 1) \, dx = \int (2x \cdot x^2 + 2x \cdot 1) \, dx = \int (2x^3 + 2x) \, dx$

Step 2: Integrate Term by Term

Now, we integrate each term separately:

1. Integrate $2x^3$: $\int 2x^3 \, dx = \frac{2x^4}{4} = \frac{x^4}{2}$

2. Integrate $2x$: $\int 2x \, dx = x^2$

Step 3: Combine the Results

Now, combine the results of the integration:

$\int 2x(x^2 + 1) \, dx = \frac{x^4}{2} + x^2 + C$

where $C$ is the constant of integration.

Final Answer:

$\int 2x(x^2 + 1) \, dx = \frac{x^4}{2} + x^2 + C$

Would you like more details on this problem or have any questions?

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Here are 5 related questions to further explore the topic:

1. How would you solve the integral $\int 3x^2(x^2 + 2) \, dx$?
2. Can you compute $\int 4x(x^2 + 5x + 6) \, dx$?
3. What is the integral of $\int (x^3 + 2x^2 + x) \, dx$?
4. How would you solve the definite integral $\int_{0}^{1} 2x(x^2 + 1) \, dx$?
5. Can you evaluate $\int x^2 \ln(x) \, dx$?

Tip: When dealing with polynomial expressions inside an integral, expanding them first can make integration much simpler.