Evaluate the integrals in Exercises 31–78.
75. ∫(from -2 to -1)2dv/(v²+4v+5)

Verified step by step guidance

1. Start by examining the integral \( \int_{-2}^{-1} \frac{2}{v^{2} + 4v + 5} \, dv \). Notice that the denominator is a quadratic expression, so the first step is to complete the square for the quadratic \( v^{2} + 4v + 5 \).
2. Rewrite the quadratic by completing the square: \( v^{2} + 4v + 5 = (v^{2} + 4v + 4) + 1 = (v + 2)^{2} + 1 \). This simplifies the integral to \( \int_{-2}^{-1} \frac{2}{(v + 2)^{2} + 1} \, dv \).
3. Make a substitution to simplify the integral further. Let \( u = v + 2 \), which implies \( du = dv \). Change the limits of integration accordingly: when \( v = -2 \), \( u = 0 \); when \( v = -1 \), \( u = 1 \). The integral becomes \( \int_{0}^{1} \frac{2}{u^{2} + 1} \, du \).
4. Recognize that the integral \( \int \frac{1}{u^{2} + 1} \, du \) corresponds to the inverse tangent function \( \arctan(u) \). Therefore, the integral can be expressed as \( 2 \int_{0}^{1} \frac{1}{u^{2} + 1} \, du = 2 [ \arctan(u) ]_{0}^{1} \).
5. Evaluate the definite integral by substituting the limits into the inverse tangent function: \( 2 ( \arctan(1) - \arctan(0) ) \). This expression represents the value of the original integral.

Key Concepts

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 definite integrals involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in the form (x + p)² + q. This simplifies integration, especially when dealing with quadratic denominators, by transforming the expression into a form that matches standard integral formulas involving arctangent or logarithmic functions.

Integration of Rational Functions

Integrating rational functions often requires algebraic manipulation such as partial fraction decomposition or substitution. When the denominator is a quadratic that cannot be factored easily, rewriting it by completing the square helps to apply standard integral formulas, like those involving arctan for expressions of the form 1/(x² + a²).