To solve the integral

$\int \frac{x^2}{\sqrt{x^2 + 5}} \, dx$

we can use a substitution method. Here's the step-by-step process:

Step 1: Substitution

Let’s set: $u = x^2 + 5$ Now, differentiate both sides with respect to $x$: $du = 2x \, dx$

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

Now, rewrite the original integral in terms of $u$:

$\int \frac{x^2}{\sqrt{x^2 + 5}} \, dx = \int \frac{(u - 5)}{\sqrt{u}} \cdot \frac{du}{2x}$

This approach isn't simplifying easily. Let's try to apply integration by parts instead or look into trigonometric substitution