Find the following integrals :
`int (2x^2+e^x) dx`

Answer

To solve the integral \( \int (2x^2 + e^x) \, dx \), we can break it down into two separate integrals:

\[ \int (2x^2 + e^x) \, dx = \int 2x^2 \, dx + \int e^x \, dx \]

### Step 1: Integrate \( 2x^2 \)

Using the power rule of integration, which states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), we can integrate \( 2x^2 \):

\[ \int 2x^2 \, dx = 2 \cdot \frac{x^{2+1}}{2+1} = 2 \cdot \frac{x^3}{3} = \frac{2}{3} x^3 \]

### Step 2: Integrate \( e^x \)

The integral of \( e^x \) is straightforward:

\[ \int e^x \, dx = e^x \]

### Step 3: Combine the results

Now, we can combine the results of the two integrals:

\[ \int (2x^2 + e^x) \, dx = \frac{2}{3} x^3 + e^x + C \]

where \( C \) is the constant of integration.

### Final Answer:

Thus, the final result of the integral is:

\[ \int (2x^2 + e^x) \, dx = \frac{2}{3} x^3 + e^x + C \]

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