It seems like you're trying to solve the integral of the expression $\int \frac{x}{\sqrt{x^2 + 1}} \, dx$, and you're providing the substitution $u = x^2 + 1$.

Let's proceed with solving this step-by-step.

Step 1: Substitution

Given $u = x^2 + 1$, we can differentiate both sides with respect to $x$ to find $du$:

$\frac{du}{dx} = 2x \quad \Rightarrow \quad du = 2x \, dx$

Thus, we can rewrite $x \, dx$ as:

$x \, dx = \frac{du}{2}$

Step 2: Substituting into the Integral

The original integral becomes:

$\int \frac{x}{\sqrt{x^2 + 1}} \, dx = \int \frac{1}{\sqrt{u}} \cdot \frac{du}{2}$

This simplifies to:

$\frac{1}{2} \int u^{-1/2} \, du$

Step 3: Solving the Integral

Now, integrate $u^{-1/2}$:

$\int u^{-1/2} \, du = 2u^{1/2}$

Thus, we get:

$\frac{1}{2} \cdot 2u^{1/2} = u^{1/2}$

Step 4: Back Substitution

Finally, substitute $u = x^2 + 1$ back into the expression:

$u^{1/2} = \sqrt{x^2 + 1}$

So, the final result is:

$\int \frac{x}{\sqrt{x^2 + 1}} \, dx = \sqrt{x^2 + 1} + C$

Where $C$ is the constant of integration.

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Would you like further explanation or a breakdown of a specific step? Here are some related questions:

1. What is the general process for solving integrals using substitution?
2. How can you recognize when to use substitution in an integral?
3. Can this method be used for other types of integrals, like rational functions?
4. What is the role of the constant of integration in indefinite integrals?
5. How does the chain rule relate to the substitution method in integration?

Tip: If you're unsure about when to use substitution, look for patterns where the integral has a composite function (like a square root) that suggests a simpler substitution.