Evaluate the integrals in Exercises 77–90.
84. ∫(from 2 to 4)2dx/(x²-6x+10)

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First, recognize that the integral is of the form \(\int_{2}^{4} \frac{2}{x^{2} - 6x + 10} \, dx\). The denominator is a quadratic expression, so start by completing the square for the quadratic \(x^{2} - 6x + 10\).

Rewrite the quadratic as \(x^{2} - 6x + 10 = (x^{2} - 6x + 9) + 1 = (x - 3)^{2} + 1\). This simplifies the integral to \(\int_{2}^{4} \frac{2}{(x - 3)^{2} + 1} \, dx\).

Make a substitution to simplify the integral: let \(u = x - 3\), so that \(du = dx\). Change the limits of integration accordingly: when \(x = 2\), \(u = 2 - 3 = -1\); when \(x = 4\), \(u = 4 - 3 = 1\).

Rewrite the integral in terms of \(u\): \(\int_{-1}^{1} \frac{2}{u^{2} + 1} \, du\). Recognize that the integral of \(\frac{1}{u^{2} + 1}\) is \(\arctan(u)\).

Use the integral formula \(\int \frac{1}{u^{2} + 1} \, du = \arctan(u) + C\) to write the integral as \(2 \int_{-1}^{1} \frac{1}{u^{2} + 1} \, du = 2 [\arctan(u)]_{-1}^{1}\). Evaluate this expression by substituting the limits.

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Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫ from a to b of f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Integration of Rational Functions

Integrating rational functions often requires algebraic manipulation such as completing the square or partial fraction decomposition. Recognizing the form of the denominator helps in choosing the right technique to simplify the integral for easier evaluation.

Completing the Square

Completing the square transforms a quadratic expression into a perfect square plus or minus a constant. This technique simplifies the denominator in integrals, making it easier to apply standard integral formulas involving inverse trigonometric functions or logarithms.

Evaluate the integrals in Exercises 77–90.84. ∫(from 2 to 4)2dx/(x²-6x+10)