The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dx / ((x - 2)√(x² - 4x + 3)))

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Start by simplifying the expression under the square root. Notice that $x^2 - 4x + 3$ can be factored or completed the square. Rewrite it as $x^2 - 4x + 3 = (x - 2)^2 - 1$.

Make the substitution $u = x - 2$ to simplify the integral. Then, $du = dx$, and the integral becomes $\int \frac{du}{u \sqrt{u^2 - 1}}$.

Recognize that the integral now has the form $\int \frac{du}{u \sqrt{u^2 - 1}}$, which suggests using a trigonometric substitution. Since $\sqrt{u^2 - 1}$ appears, consider the substitution $u = \sec \theta$, so that $\sqrt{u^2 - 1} = \tan \theta$.

Express $du$ in terms of $d\theta$: since $u = \sec \theta$, then $du = \sec \theta \tan \theta \, d\theta$. Substitute these into the integral to rewrite it entirely in terms of $\theta$.

Simplify the resulting integral in $\theta$ and integrate using standard trigonometric integral formulas. After integration, substitute back $\theta = \sec^{-1}(u)$ and then $u = x - 2$ to express the answer in terms of $x$.

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Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By letting a new variable represent a part of the integrand, the integral can often be transformed into a more manageable form. This technique is especially useful when the integrand contains composite functions or expressions under radicals.

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Substitution With an Extra Variable

Simplifying Radicals and Quadratic Expressions

Simplifying the expression under the square root, often a quadratic, helps in recognizing patterns or suitable substitutions. Completing the square or factoring the quadratic can transform the radical into a standard form, making the integral easier to evaluate using known formulas or trigonometric substitutions.

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Simplifying Trig Expressions

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions involving square roots of quadratic polynomials with trigonometric functions. This method leverages identities like sin²θ + cos²θ = 1 to simplify integrals containing radicals, converting them into integrals of trigonometric functions that are easier to solve.

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