Key Concepts and Formulas

• Indefinite Integration: The process of finding the antiderivative of a function.
• Substitution Method: A technique to simplify integrals by substituting a part of the integrand with a new variable.
• $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$ and C is the constant of integration.

Step-by-Step Solution

Step 1: Perform a u-substitution. We are given the integral $\int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx$. To simplify this, we'll use the substitution method. Let $t = \sqrt{2x-1}$. This substitution is chosen because it simplifies the square root in the denominator.

Step 2: Solve for x and find dx in terms of dt. Square both sides of the substitution equation to get $t^2 = 2x - 1$. Now, solve for x: $2x = t^2 + 1 \Rightarrow x = \frac{t^2 + 1}{2}$. Next, differentiate both sides of $t^2 = 2x - 1$ with respect to x: $2t \frac{dt}{dx} = 2$. This implies $dx = t \, dt$.

Step 3: Substitute into the integral. Substitute $x = \frac{t^2 + 1}{2}$ and $dx = t \, dt$ into the original integral: $\int {{{x + 1} \over {\sqrt {2x - 1} }}} \,dx = \int {{{\frac{t^2 + 1}{2} + 1} \over t}} \, t \, dt = \int {{\frac{t^2 + 1 + 2}{2}}} \, dt = \int {{\frac{t^2 + 3}{2}}} \, dt$.

Step 4: Evaluate the simplified integral. Now, integrate with respect to t: $\int {{\frac{t^2 + 3}{2}}} \, dt = \frac{1}{2} \int (t^2 + 3) \, dt = \frac{1}{2} \left( \frac{t^3}{3} + 3t \right) + C = \frac{t^3}{6} + \frac{3t}{2} + C$.

Step 5: Factor out t and simplify. Factor out t from the expression: $\frac{t^3}{6} + \frac{3t}{2} + C = \frac{t}{6} (t^2 + 9) + C$.

Step 6: Substitute back for t. Substitute $t = \sqrt{2x-1}$ back into the expression: $\frac{\sqrt{2x-1}}{6} ((\sqrt{2x-1})^2 + 9) + C = \frac{\sqrt{2x-1}}{6} (2x - 1 + 9) + C = \frac{\sqrt{2x-1}}{6} (2x + 8) + C$.

Step 7: Simplify the expression. $\frac{\sqrt{2x-1}}{6} (2x + 8) + C = \frac{\sqrt{2x-1}}{3} (x + 4) + C$.

Step 8: Identify f(x). We are given that the integral is equal to $f(x) \sqrt{2x-1} + C$. Comparing this with our result, we can identify $f(x) = \frac{x+4}{3}$.

Step 9: Rewrite f(x) to match the options. $f(x) = \frac{1}{3}(x+4) = \frac{1}{3}x + \frac{4}{3}$. The correct answer is $\frac{1}{3}(x+4)$. This corresponds to option (B).

Common Mistakes & Tips

• Forgetting the constant of integration: Always remember to add the constant of integration, C, when evaluating indefinite integrals.
• Incorrectly substituting back: Be careful when substituting back to the original variable. Ensure you have correctly solved for x in terms of t and substitute the right expression.
• Algebraic errors: Double-check your algebraic manipulations to avoid mistakes.

Summary

We used the substitution method to evaluate the given indefinite integral. By substituting $t = \sqrt{2x-1}$, we simplified the integral and were able to find its antiderivative. After substituting back to the original variable x, we identified the function f(x) as $\frac{x+4}{3}$, which matches option (B).

Final Answer The final answer is \boxed{\frac{1}{3}(x+4)}, which corresponds to option (B).