$$ \int\frac{x}{\sqrt{4+x^2}} $$$$ \differentialD x $$

The integral you provided is:

\[
I = \int \frac{x}{\sqrt{4+x^2}} \, dx
\]

We can solve this by using a substitution. Let's set:

\[
u = 4 + x^2
\]

Then, differentiating $u$ with respect to $x$:

\[
du = 2x \, dx
\]

This means:

\[
\frac{du}{2} = x \, dx
\]

Now, we can rewrite the integral in terms of $u$:

\[
I = \int \frac{x}{\sqrt{4+x^2}} \, dx = \int \frac{1}{2} \frac{du}{\sqrt{u}}
\]

This simplifies to:

\[
I = \frac{1}{2} \int u^{-\frac{1}{2}} \, du
\]

The integral of $ u^{-\frac{1}{2}} $ is:

\[
\int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}} = 2\sqrt{u}
\]

Substituting back $ u = 4 + x^2 $, we get:

\[
I = \frac{1}{2} \cdot 2\sqrt{4 + x^2} = \sqrt{4 + x^2}
\]

Thus, the solution to the integral is:

\[
I = \sqrt{4 + x^2} + C
\]

where $C$ is the constant of integration.

### Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you approach integrals involving trigonometric substitutions?
2. What are some examples of integrals that require partial fraction decomposition?
3. How do you solve integrals involving rational functions and radicals?
4. Can you explain the method of integration by parts with an example?
5. How can substitution help in solving definite integrals?

**Tip:** Substitution is a powerful tool for simplifying integrals, especially when the integrand involves compositions of functions.

This page provides a detailed solution to the integral $$ \int \frac{x}{\sqrt{4+x^2}} \, dx $$ using the substitution method. Learn the key steps, including variable substitution and solving integrals with radicals, suited for calculus students.

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