Evaluate the integrals in Exercises 77–90.
81. ∫dy/(y²-2y+5)

Verified step by step guidance

1. Start by rewriting the quadratic expression in the denominator to a completed square form. The denominator is \(y^2 - 2y + 5\). Complete the square by expressing it as \((y - a)^2 + b\) for some constants \(a\) and \(b\).
2. Recall that to complete the square for \(y^2 - 2y + 5\), you take half of the coefficient of \(y\), which is \(-2\), divide by 2 to get \(-1\), then square it to get \(1\). So rewrite the denominator as \((y - 1)^2 + (5 - 1)\).
3. Simplify the expression inside the parentheses to get \((y - 1)^2 + 4\). Now the integral becomes \(\int \frac{dy}{(y - 1)^2 + 4}\).
4. Recognize that this integral matches the standard form \(\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C\). Here, \(x = y - 1\) and \(a^2 = 4\), so \(a = 2\).
5. Make the substitution \(u = y - 1\), so \(du = dy\). Rewrite the integral in terms of \(u\) and apply the formula to express the integral in terms of the arctangent function.

Key Concepts

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in the form (y - h)² + k. This simplifies the integral by transforming the denominator into a recognizable form, often enabling the use of standard integral formulas involving inverse trigonometric functions.

Integration of Rational Functions

Integrating rational functions involves expressing the integrand in a simpler form, often by partial fraction decomposition or algebraic manipulation. For quadratics that do not factor easily, rewriting the denominator helps apply known integral formulas.

Inverse Trigonometric Integrals

Certain integrals involving expressions like 1/(x² + a²) correspond to inverse trigonometric functions such as arctangent. Recognizing these forms allows direct application of formulas like ∫dx/(x² + a²) = (1/a) arctan(x/a) + C.