We are tasked with finding the integral of the function \( x \cdot \sin(x^2 + 1) \). 

The best approach here is to use substitution. Let us proceed with the solution:

### Step 1: Substitution
Let:
\[
u = x^2 + 1
\]
Then, differentiate both sides with respect to \( x \):
\[
\frac{du}{dx} = 2x
\]
This implies:
\[
du = 2x \, dx \quad \text{or} \quad \frac{du}{2} = x \, dx
\]

### Step 2: Substitute into the integral
Now, substitute \( u \) and \( du \) into the original integral:
\[
\int x \sin(x^2 + 1) \, dx = \int \sin(u) \cdot \frac{du}{2}
\]

### Step 3: Solve the integral
The integral of \( \sin(u) \) is \( -\cos(u) \). Therefore:
\[
\int \sin(u) \, \frac{du}{2} = -\frac{1}{2} \cos(u) + C
\]

### Step 4: Back-substitute \( u = x^2 + 1 \)
Now, replace \( u \) with \( x^2 + 1 \):
\[
-\frac{1}{2} \cos(x^2 + 1) + C
\]

Thus, the integral of \( x \cdot \sin(x^2 + 1) \) is:
\[
\boxed{-\frac{1}{2} \cos(x^2 + 1) + C}
\]

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you approach integrals involving products of functions?
2. What is the general method of substitution in integrals?
3. Can integration by parts be applied to similar problems?
4. How do we verify the result of an integral?
5. What are the properties of trigonometric integrals?

**Tip**: Always try substitution when you see a composition of functions inside another function, as it simplifies many complex integrals.

Learn how to solve the integral of x · sin(x² + 1) with a step-by-step solution using substitution. This problem demonstrates key calculus techniques such as substitution and trigonometric integration. Suitable for undergraduate students.

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