Evaluate the integrals. ∫(e^x sin^-1 e^x)/(√(1-e^2 x)) d x |...

Name: Problem 90, Chapter 7: Thomas Calculus in SI Units
Uploaded: 2023-06-02T02:20:23-08:00
Duration: 2 min
Channel: Linda Hand
Description: Evaluate the integrals. ∫(e^x sin^-1 e^x)/(√(1-e^2 x)) d x

Key Concepts

Derivative of Inverse Trigonometric Functions

Understanding that the derivative of sin?¹(u) is 1/?(1 - u²) (when u is a function of x, multiplied by u') is crucial. In this problem, recognizing that the derivative of sin?¹(e?) gives rise to the expression e?/?(1 - e^(2x)) helps simplify the integral by revealing an inherent derivative structure.

Chain Rule

The chain rule is key when dealing with composite functions. In the context of this integral, it explains how the derivative of sin?¹(e?) naturally produces the product of e? and the derivative of sin?¹ with respect to its argument, thereby linking the exponential and trigonometric parts of the integrand.

Integration by Recognition

Integration by recognition involves identifying that an integrand is the derivative (or a scalar multiple) of a known composite function. Here, noting that the derivative of [sin?¹(e?)]² is 2 sin?¹(e?) multiplied by e?/?(1 - e^(2x)) allows one to immediately relate the given integral to an antiderivative of the squared function, simplifying the integration process.

Substitution Method

The substitution method is a general integration technique that simplifies an integral by replacing a complicated expression with a single variable. In problems like this, substituting u = e? or recognizing u = sin?¹(e?) can help transform the integral into a more familiar form and make the application of other integration techniques more straightforward.

Transcript

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