The given integral is:

$\int \frac{dx}{\sqrt{x+1}}$

We are asked to solve this using a change of variables.

Step-by-Step Solution:

1. Substitution: Let's set $u = x + 1$, so $du = dx$.

   The integral becomes: $\int \frac{du}{\sqrt{u}} = \int u^{-\frac{1}{2}} du$

2. Integration: Now, integrate $u^{-\frac{1}{2}}$: $\int u^{-\frac{1}{2}} du = 2u^{\frac{1}{2}} + C$

3. Substitute back $u = x + 1$: $2\sqrt{x+1} + C$

Thus, the solution to the integral is $2\sqrt{x+1} + C$.

Correct Option:

Looking at the choices given in the image, the correct answer is option A.

Would you like more details or have any questions about this process?

Here are 5 related questions to deepen your understanding:

1. How do you choose an appropriate substitution in integrals?
2. What other methods can be used to solve similar integrals?
3. Can this integral be solved without substitution? If so, how?
4. How would the solution change if the integrand had a different constant added to $x$?
5. What is the significance of the constant $C$ in indefinite integrals?

Tip: Always check if a substitution simplifies the integrand significantly; it’s a powerful technique for solving complex integrals!