integrate wrt to x dx/squareroot x^2-5x+7

To evaluate the integral \(\int \frac{dx}{\sqrt{x^2 - 5x + 7}}\), we can solve it using a standard substitution method or by recognizing it as a form that leads to an inverse hyperbolic function.

### Step 1: Complete the square
First, we need to complete the square for the quadratic expression under the square root:

\[
x^2 - 5x + 7 = \left(x^2 - 5x + \frac{25}{4}\right) + 7 - \frac{25}{4}
\]
\[
= \left(x - \frac{5}{2}\right)^2 - \frac{3}{4}
\]

So, the integral becomes:

\[
\int \frac{dx}{\sqrt{\left(x - \frac{5}{2}\right)^2 - \frac{3}{4}}}
\]

### Step 2: Use a trigonometric substitution
Let \( u = x - \frac{5}{2} \), so \( du = dx \). The integral is now:

\[
\int \frac{du}{\sqrt{u^2 - \left(\frac{\sqrt{3}}{2}\right

Learn how to evaluate the integral ∫ dx/√(x^2 - 5x + 7) using advanced integration techniques. This problem involves substitution methods and completing the square, crucial in integral calculus. Suitable for advanced high school students and undergraduates.

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