The integral given is:

$\int \frac{1}{2x + 5} \, dx$

To solve this integral, we can use a substitution method.

Solution

1. Let $u = 2x + 5$.

   • Then, $\frac{du}{dx} = 2$ or $dx = \frac{du}{2}$.

2. Substitute $u$ and $dx$ in the integral: $\int \frac{1}{2x + 5} \, dx = \int \frac{1}{u} \cdot \frac{du}{2}$

3. Simplify the integral: $= \frac{1}{2} \int \frac{1}{u} \, du$

4. Now integrate: $= \frac{1}{2} \ln |u| + C$

5. Substitute back $u = 2x + 5$: $= \frac{1}{2} \ln |2x + 5| + C$

Final Answer

$\int \frac{1}{2x + 5} \, dx = \frac{1}{2} \ln |2x + 5| + C$

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

Here are some related questions for further practice:

1. How would the integral change if the denominator were $3x + 5$ instead of $2x + 5$?
2. How do we handle integrals of functions like $\int \frac{1}{ax + b} \, dx$ in general?
3. What is the role of substitution in solving integrals, and why is it useful?
4. How would the solution differ if the integral were $\int \frac{1}{x^2 + 5} \, dx$ instead?
5. What would be the steps to solve $\int \frac{2}{2x + 5} \, dx$?

Tip: When dealing with integrals of rational functions, look for substitution opportunities to simplify the expression.