Question

$67-84$ Evaluate the integral. $$\int e^{\cos t} \sin t d t$$

   $67-84$ Evaluate the integral. 
$$\int e^{\cos t} \sin t d t$$

Calculus

James Stewart 8th Edition

Chapter 4, Problem 72

Instant Answer

Solved by Expert Priyanka Sadarangani

03/13/2021

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Let $u = \cos t$. Show more…

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$67-84$ Evaluate the integral. $$\int e^{\cos t} \sin t d t$$

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Key Concepts

U-Substitution

This technique simplifies integration by replacing a part of the integrand with a new variable. By selecting a substitution that matches a component of the derivative present in the integrand, the integral can be transformed into a basic form that is easier to evaluate.

Reverse Chain Rule

Also known as the method of undoing the chain rule, this concept is used when the derivative of a function inside another function appears in the integrand. Recognizing this pattern allows one to apply the reverse chain rule through substitution, facilitating the integration of composite functions.

Integration of Exponential Functions

Exponential functions, particularly those of the form e^u, have the property of being their own derivative. This characteristic makes them straightforward to integrate, especially when combined with substitution, since the antiderivative of e^u is simply e^u plus a constant of integration.

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