The integral you've uploaded is:

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

Solution:

To solve this integral, we can use substitution. Let's define:

$u = x^2 + 1$

Then, differentiating both sides with respect to $x$:

$du = 2x \, dx$

Thus, $x \, dx = \frac{1}{2} \, du$.

Substitute into the integral:

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

The integral of $\frac{1}{\sqrt{u}}$ is $2\sqrt{u}$, so we get:

$\frac{1}{2} \cdot 2\sqrt{u} = \sqrt{u}$

Now, substitute back $u = x^2 + 1$:

$\sqrt{x^2 + 1} + C$

where $C$ is the constant of integration.

Final Answer:

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

Would you like further details, or do you have any questions on this process?

5 Related Questions:

1. How can substitution be used to simplify other integrals?
2. What other types of integrals are solved using the substitution method?
3. How does the choice of substitution affect the difficulty of solving an integral?
4. Could integration by parts be used to solve a similar integral?
5. What are the common forms of integrals involving square roots?

Tip:

When choosing a substitution, try to express complex parts of the integral in terms of a single variable to simplify the expression.