`int2xsin(x^(2)+1)dx`

Explanation

To solve the integral \( \int 2x \sin(x^2 + 1) \, dx \), we will use the method of substitution. Here are the steps:

### Step 1: Choose a substitution
Let \( t = x^2 + 1 \).

### Step 2: Differentiate the substitution
Now, we differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = 2x \]
This implies that:
\[ dt = 2x \, dx \]

### Step 3: Rewrite the integral
Now we can rewrite the integral in terms of \( t \):
\[ \int 2x \sin(x^2 + 1) \, dx = \int \sin(t) \, dt \]

### Step 4: Integrate
The integral of \( \sin(t) \) is:
\[ -\cos(t) + C \]

### Step 5: Substitute back
Now we substitute back \( t = x^2 + 1 \):
\[ -\cos(x^2 + 1) + C \]

### Final Answer
Thus, the final result of the integral is:
\[ \int 2x \sin(x^2 + 1) \, dx = -\cos(x^2 + 1) + C \]

---