The expression you provided from the image is:

$\int \sqrt{x + \frac{5^2}{4}} \, dx$

This integral can be simplified by recognizing that $\frac{5^2}{4} = \frac{25}{4}$. Therefore, the expression under the square root becomes:

$\int \sqrt{x + \frac{25}{4}} \, dx$

To solve this integral, we can use the substitution method. Let’s introduce a new variable to simplify the expression:

1. Set $u = x + \frac{25}{4}$, so that $du = dx$.
2. The integral then becomes:

$\int \sqrt{u} \, du$

This is a standard integral. The antiderivative of $\sqrt{u}$ is:

$\frac{2}{3} u^{3/2} + C$

Finally, substitute back $u = x + \frac{25}{4}$:

$\frac{2}{3} \left( x + \frac{25}{4} \right)^{3/2} + C$

Thus, the solution to the integral is:

$\frac{2}{3} \left( x + \frac{25}{4} \right)^{3/2} + C$

Would you like further details on any step, or do you have any additional questions?

Related Questions:

1. How does substitution work in integrals?
2. What are the common techniques for solving integrals involving square roots?
3. How do you determine the correct substitution for simplifying an integral?
4. How can this result be applied to real-world problems?
5. What other types of integrals involve similar patterns?

Tip:

Always check if you can simplify or factor the expression inside a square root before attempting to integrate, as it can often make the process much easier!