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The integral that we need to solve is:

\[
\int \frac{dx}{x^2 - x + 2}
\]

### Step 1: Completing the square

The denominator \( x^2 - x + 2 \) is a quadratic expression. To make it easier to integrate, we first complete the square:

\[
x^2 - x + 2 = \left( x - \frac{1}{2} \right)^2 + \frac{7}{4}
\]

This form will help us recognize a standard integral.

### Step 2: Substitute to simplify the integral

Let’s use the substitution:

\[
u = x - \frac{1}{2}, \quad du = dx
\]

Now, the integral becomes:

\[
\int \frac{du}{u^2 + \frac{7}{4}}
\]

We can simplify the denominator:

\[
u^2 + \frac{7}{4} = \frac{4u^2 + 7}{4}
\]

Thus, the integral is now:

\[
\int \frac{4 \, du}{4u^2 + 7}
\]

### Step 3: Standard integral form

Next, recognize that the integral is in a standard form. Recall the standard formula:

\[
\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C
\]

In our case, \( 4u^2 + 7 \) can be written as \( \left(2u\right)^2 + \left(\sqrt{7}\right)^2 \), and we can apply the formula for this specific form:

\[
\int \frac{4 \, du}{4u^2 + 7} = \frac{4}{\sqrt{7}} \tan^{-1}\left(\frac{2u}{\sqrt{7}}\right) + C
\]

### Step 4: Substituting back \( u = x - \frac{1}{2} \)

Now that we've solved the integral in terms of \( u \), substitute back \( u = x - \frac{1}{2} \) to get the final result:

\[
\int \frac{dx}{x^2 - x + 2} = \frac{4}{\sqrt{7}} \tan^{-1}\left(\frac{2\left(x - \frac{1}{2}\right)}{\sqrt{7}}\right) + C
\]

### Final Answer:

\[
\boxed{\frac{4}{\sqrt{7}} \tan^{-1}\left(\frac{2(x - \frac{1}{2})}{\sqrt{7}}\right) + C}
\]

Would you like any further clarification or details? 

Here are 5 questions that expand on this topic:
1. How do you handle integrals of higher-order polynomials in the denominator?
2. What are the applications of inverse trigonometric integrals like \( \tan^{-1} \)?
3. How can you integrate rational functions where the denominator is factored into linear terms?
4. How does completing the square help in solving integrals involving quadratic expressions?
5. What is the general strategy for solving integrals that result in standard inverse trigonometric forms?

Tip: When completing the square, always check that the constant added or subtracted results in a perfect square trinomial.

Evaluate the indefinite integral \( \int \frac{dx}{x^2 - x + 2} \)

Learn how to evaluate the indefinite integral \( \int \frac{dx}{x^2 - x + 2} \) step-by-step, including completing the square, substitution, and using inverse trigonometric formulas. Suitable for advanced calculus students in Grades 11-12.

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