Evaluate the integral. (a) ∫5^-2 x d x (b) ∫1^2 5^-2 x d x |...

Name: Problem 29, Chapter 7: Calculus
Uploaded: 2022-03-02T18:11:36-08:00
Duration: 2 min 15 s
Channel: Gregory Higby
Description: Evaluate the integral. (a) ∫5^-2 x d x (b) ∫1^2 5^-2 x d x

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Key Concepts

Exponential Function Integration

This concept involves integrating functions where the variable appears in the exponent with a constant base. A key result is that the integral of an exponential function of the form a^(kx) can be computed using the formula ?a^(kx)dx = a^(kx)/(k ln(a)) + C, where a is a positive constant not equal to 1, and k is a constant. This formula is essential when dealing with integrals such as that of 5^(-2x).

Substitution Method in Integration

When the exponent of an exponential function includes a linear expression, the substitution method (or u-substitution) is utilized to simplify the integral. By letting u equal the linear expression in the exponent, the differential dx is correspondingly adjusted. This technique helps in reducing the integral to a standard form that is easier to integrate.

Definite Integration and the Fundamental Theorem of Calculus

Definite integration involves evaluating the antiderivative of a function over a specified interval. The Fundamental Theorem of Calculus states that if F is an antiderivative of f, then the definite integral from a to b is given by F(b) - F(a). This concept is crucial for calculating the exact numerical value of an integral over a given interval, as illustrated by evaluating the integral of 5^(-2x) from x = 1 to x = 2.

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