5-32 Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. ∫(d x)/(√(1-e^

5-32 Use the Table of Integrals on Reference Pages 6-10 to...

Key Concepts

Reference to Standard Integral Tables

Using standard integral tables streamlines the integration process by providing known results for common integrals. These tables include a variety of antiderivatives, including those that result from substitutions and trigonometric integrations, allowing one to efficiently match an integral to a listed formula and thus find its antiderivative.

Inverse Trigonometric Integration

Many integrals that arise after a trigonometric substitution reduce to standard forms involving inverse trigonometric functions. In these cases, tables of integrals often list results that involve inverse sine, cosine, or tangent functions. Recognizing these forms and applying the corresponding formulas is essential for evaluating the original integral.

Trigonometric Substitution

Trigonometric substitution is a specific type of substitution used when the integrand contains expressions under a square root that resemble quadratic forms. By setting a variable equal to a trigonometric function (such as sine or cosine), the square root simplifies using trigonometric identities, allowing the integral to be expressed in terms of standard trigonometric integrals.

Substitution Method

This technique involves replacing a complicated expression with a new variable to simplify the integrand. In integration, an appropriate substitution can transform an integral that looks difficult into one that fits a standard form, making it easier to evaluate with known formulas or techniques.

Transcript

00:01 All right, for this problem, the key concept is integration using tables, and the first steps for those problems are making substitutions and finding the correct entry on the table.

00:12 Over here in red, this is the entry we're going to end up using.

00:16 I'll explain why in just a minute.

00:18 But the problem we're going to start with today is the integral of dx over the square root of 1 minus e to the 2x.

00:27 And you can see i've rewritten that here as the integral of dx over the square root of 1 squared minus e to the x squared, using some exponent rules here on the e to the x part.

00:40 And so this has the form, the square root of a squared minus u squared.

00:46 And so i would start looking on the table for things of that form, right, in the denominator.

00:54 But if you notice up here, there's an extra u out here that i don't have.

00:57 And we'll see why that is in just a minute.

01:00 That deals with the substitutions we have to make.

01:08 Okay, so let's get our substitutions made here.

01:13 We're going to let u be e to the x.

01:18 So we'll let u equal e to the x.

01:23 And then my du, well, the derivative of e to the x is e to the x.

01:29 So du is e to the x, dx.

01:32 So dividing to find dx, i get dx is du over e to the x.

01:41 So let's do our substitutions.