For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx

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Start by completing the square for the expression inside the square root: \(x^{2} - 2x + 5\). To do this, rewrite it as \(\left(x^{2} - 2x + 1\right) + 4\) because \(1\) is the square of half the coefficient of \(x\).

Express the completed square form explicitly: \(\left(x - 1\right)^{2} + 4\). So the integral becomes \(\int \frac{1}{\sqrt{\left(x - 1\right)^{2} + 4}} \, dx\).

Recognize that the integral now has the form \(\int \frac{1}{\sqrt{u^{2} + a^{2}}} \, du\) where \(u = x - 1\) and \(a = 2\). This suggests using the trigonometric substitution \(u = a \tan \theta\), or \(x - 1 = 2 \tan \theta\).

Differentiate the substitution to find \(dx\): since \(x - 1 = 2 \tan \theta\), then \(dx = 2 \sec^{2} \theta \, d\theta\).

Rewrite the integral in terms of \(\theta\) using the substitution and simplify the square root using the identity \(1 + \tan^{2} \theta = \sec^{2} \theta\). Then proceed to integrate with respect to \(\theta\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Completing the Square

Completing the square is a method used to rewrite a quadratic expression in the form ax² + bx + c as a perfect square plus or minus a constant. This technique simplifies the integrand, making it easier to identify the appropriate substitution, especially for integrals involving square roots of quadratic expressions.

Trigonometric Substitution

Trigonometric substitution is a technique used to evaluate integrals involving square roots of quadratic expressions by substituting variables with trigonometric functions. Depending on the form (x² ± a² or a² - x²), substitutions like x = a sec θ or x = a tan θ transform the integral into a trigonometric integral that is easier to solve.

Integration of Functions Involving Square Roots

Integrals containing square roots of quadratic expressions often require algebraic manipulation and substitution to simplify. Understanding how to handle these integrals involves recognizing patterns, applying completing the square, and using trigonometric identities to rewrite and integrate the function effectively.

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