Evaluate $\int (2x + 3)^{\frac{1}{2}} \delta x$

To solve the integral $\int (2x + 3)^{\frac{1}{2}} \, dx$, we use the method of Integration by Substitution (or the reverse chain rule).

1. Analyze the Concept

When integrating a function of the form $(ax + b)^n$, the rule is:

Where:

• $a$ is the coefficient of $x$ (in this case, $2$).
• $n$ is the power (in this case, $\frac{1}{2}$).
• $k$ is the constant of integration.

2. Step-by-Step Solution

Step 1: Identify the components.

• $n = \frac{1}{2}$
• $a = 2$

Step 2: Apply the integration formula. Increase the power by 1:

Divide by the new power and multiply by the derivative of the inner function (which is $2$):

Step 3: Simplify the expression. The $2$ in the denominator and the $2$ in the denominator of the fraction $\frac{3}{2}$ cancel out:

So, the expression becomes:

Which can be written as:

3. Conclusion

Comparing our result to the given options, we find that it matches option C.

Exam Tip: Whenever you see a linear expression inside a power (like $2x+3$), always remember to divide by the coefficient of $x$. Many students forget to divide by the $2$, which leads to common errors. Always check if your answer differentiates back to the original function to verify your work!