Question

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

Accepted Answer

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.

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

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Key Concepts

Definite Integrals

Integration of Rational Functions

Completing the Square