Evaluate the integrals in Exercises 31–78.
33. ∫e^x sec²(e^x - 7)dx

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Identify the integral to solve: \(\int e^x \sec^2(e^x - 7) \, dx\).

Recognize that the integrand contains a composite function \(\sec^2(u)\) where \(u = e^x - 7\), and the derivative of \(u\) with respect to \(x\) is \(\frac{du}{dx} = e^x\).

Use substitution by letting \(u = e^x - 7\). Then, compute \(du = e^x \, dx\), which means \(e^x \, dx = du\).

Rewrite the integral in terms of \(u\): \(\int \sec^2(u) \, du\).

Recall that the integral of \(\sec^2(u)\) with respect to \(u\) is \(\tan(u) + C\). After integrating, substitute back \(u = e^x - 7\) to express the answer in terms of \(x\).

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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, differentiating it, and rewriting the integral in terms of this variable to make it easier to solve.

Derivative of the Exponential Function

The exponential function e^x has the unique property that its derivative is itself, e^x. This fact is crucial when differentiating or integrating expressions involving e^x, as it often simplifies the process of substitution or integration.

Integration of Secant Squared Function

The integral of sec²(u) with respect to u is tan(u) + C. Recognizing sec² as the derivative of tan allows for straightforward integration when the integrand includes sec² of a function, especially after applying substitution.

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