Key Concepts and Formulas

• Indefinite Integrals: The process of finding a function whose derivative is a given function.
• Substitution Method: A technique used to simplify integrals by substituting a part of the integrand with a new variable.
• Derivative of $\sin(x)$: $\frac{d}{dx} \sin(x) = \cos(x)$.

Step-by-Step Solution

Step 1: Rewrite the integral

We begin by rewriting the given integral to make it easier to work with: $\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {4{x^2} - 4x + 6} }}} dx = \int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {{{(2x - 1)}^2} + 5} }}} dx$ This is because $4x^2 - 4x + 6 = 4x^2 - 4x + 1 + 5 = (2x-1)^2 + 5$. This simplification is crucial for the substitution method.

Step 2: Perform a u-substitution

Let $u = \sqrt{(2x-1)^2 + 5}$. Then, $u^2 = (2x-1)^2 + 5$. We will differentiate both sides with respect to $x$ to find $du$ in terms of $dx$.

Step 3: Differentiate both sides

Differentiating $u^2 = (2x-1)^2 + 5$ with respect to $x$, we get: $2u \frac{du}{dx} = 2(2x-1)(2)$ $2u \, du = 4(2x-1) \, dx$ $u \, du = 2(2x-1) \, dx$ $\frac{1}{2} u \, du = (2x-1) \, dx$

Step 4: Substitute into the integral

Substitute $u = \sqrt{(2x-1)^2 + 5}$ and $(2x-1)dx = \frac{1}{2} u \, du$ into the integral: $\int {{{(2x - 1)\cos \sqrt {{{(2x - 1)}^2} + 5} } \over {\sqrt {{{(2x - 1)}^2} + 5} }}} dx = \int \frac{\cos(u)}{u} \cdot \frac{1}{2} u \, du = \frac{1}{2} \int \cos(u) \, du$

Step 5: Evaluate the simplified integral

The integral of $\cos(u)$ is $\sin(u)$, so we have: $\frac{1}{2} \int \cos(u) \, du = \frac{1}{2} \sin(u) + c$

Step 6: Substitute back for x

Substitute $u = \sqrt{(2x-1)^2 + 5}$ back into the expression: $\frac{1}{2} \sin(u) + c = \frac{1}{2} \sin\left(\sqrt{(2x-1)^2 + 5}\right) + c$

Common Mistakes & Tips

• Algebraic Errors: Be careful when expanding and simplifying expressions, especially when dealing with squares and square roots.
• Substitution Simplification: Choose substitutions that simplify the integral significantly. Recognizing the relationship between $(2x-1)$ and $(2x-1)^2 + 5$ is key.
• Don't Forget to Substitute Back: Remember to substitute back to the original variable after evaluating the integral.

Summary

We used the substitution method to solve the indefinite integral. By substituting $u = \sqrt{(2x-1)^2 + 5}$, we simplified the integral to $\frac{1}{2} \int \cos(u) \, du$. Evaluating this integral and substituting back gave us the final answer: $\frac{1}{2} \sin\left(\sqrt{(2x-1)^2 + 5}\right) + c$.

Final Answer

The final answer is $\boxed{{1 \over 2}\sin \sqrt {{{(2x - 1)}^2} + 5} + c}$, which corresponds to option (A).