Evaluate the integrals in Exercises 31–78.
31. ∫e^x sin(e^x)dx

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Recognize that the integral is of the form \(\int e^x \sin(e^x) \, dx\), where the argument of the sine function is \(e^x\) and the outside factor is also \(e^x\).

Use substitution by letting \(u = e^x\). Then, compute the differential \(du = e^x \, dx\), which implies \(dx = \frac{du}{e^x} = \frac{du}{u}\).

Rewrite the integral in terms of \(u\): since \(e^x \, dx = du\), the integral becomes \(\int \sin(u) \, du\).

Integrate \(\sin(u)\) with respect to \(u\), which is \(-\cos(u) + C\).

Substitute back \(u = e^x\) to express the answer in terms of \(x\): the integral is \(-\cos(e^x) + C\).

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Key Concepts

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

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function.

Chain Rule in Reverse (u-substitution)

The chain rule in differentiation helps find derivatives of composite functions. Integration by substitution is essentially the reverse process, where you recognize the inner function and its derivative within the integral. This allows rewriting the integral in terms of a single variable, facilitating easier integration.

Integration of Trigonometric Functions

Integrating trigonometric functions like sine and cosine often requires recognizing standard integral forms or applying substitution. Understanding how to integrate functions involving sine or cosine, especially when combined with other functions, is crucial for solving integrals like ∫e^x sin(e^x) dx.

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