Evaluate the integrals in Exercises 31–78.
55. ∫(from -2 to -1)e^(-(x+1)) dx

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Identify the integral to be evaluated: \(\int_{-2}^{-1} e^{-(x+1)} \, dx\).

Rewrite the exponent to clarify the integrand: \(e^{-(x+1)} = e^{-x-1} = e^{-1} \cdot e^{-x}\).

Set up a substitution to simplify the integral. Let \(u = x + 1\), then \(du = dx\). Change the limits accordingly: when \(x = -2\), \(u = -1\); when \(x = -1\), \(u = 0\).

Rewrite the integral in terms of \(u\): \(\int_{u=-1}^{0} e^{-u} \, du\).

Integrate \(e^{-u}\) with respect to \(u\): the antiderivative is \(-e^{-u}\). Then apply the new limits \(u = -1\) to \(u = 0\) to express the definite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

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. The result is a number representing the accumulated value of the function over that interval.

Integration of Exponential Functions

Integrating exponential functions like e^(ax) involves using the rule ∫ e^(ax) dx = (1/a) e^(ax) + C. When the exponent is a linear function of x, substitution may be used to simplify the integral before applying this rule.

Substitution Method

The substitution method simplifies integrals by changing variables. For an integral involving a composite function, set u equal to the inner function, then rewrite dx in terms of du. This transforms the integral into a simpler form that is easier to evaluate.

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