The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ e^(z + eᶻ) dz

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Identify the integral to solve: \(\int e^{z + e^{z}} \, dz\).

Rewrite the integrand by separating the exponent: \(e^{z + e^{z}} = e^{z} \cdot e^{e^{z}}\).

Consider a substitution to simplify the integral. Let \(u = e^{z}\), so that \(\frac{du}{dz} = e^{z} = u\), which implies \(dz = \frac{du}{u}\).

Rewrite the integral in terms of \(u\): \(\int e^{z} \cdot e^{e^{z}} \, dz = \int u \cdot e^{u} \cdot \frac{du}{u} = \int e^{u} \, du\).

Integrate \(\int e^{u} \, du\) to get \(e^{u} + C\), then substitute back \(u = e^{z}\) to express the answer in terms of \(z\).

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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.

Exponential Functions and Their Properties

Exponential functions have the form e^u, where u is a function of the variable. Understanding how to differentiate and integrate exponential functions, especially when the exponent is itself a function, is crucial. Recognizing the chain rule in reverse helps in integrating such expressions.

Recognizing Composite Functions in Integrals

Composite functions are functions within functions, such as e^(z + e^z). Identifying the inner function and its derivative within the integrand allows the use of substitution. This recognition simplifies the integral by reducing it to a basic form that is easier to evaluate.

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