Q.No.1 Solve the following integral function; [2X(X²+1)dx.

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.

Learn how to solve the integral 2x(x^2 + 1) dx with a detailed step-by-step solution. This problem involves polynomial integration, specifically integrating a product of a linear function and a quadratic function. Suitable for high school students.

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