Evaluate the integrals in Exercises 33–52.
∫ eˣ sec³(eˣ) dx

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Recognize that the integral is of the form \(\int e^{x} \sec^{3}(e^{x}) \, dx\). Notice that the argument of the secant function is \(e^{x}\), which suggests a substitution involving \(e^{x}\).

Let \(u = e^{x}\). Then, compute the differential \(du = e^{x} dx\), which implies \(du = u \, dx\) or equivalently \(dx = \frac{du}{u}\).

Rewrite the integral in terms of \(u\): substitute \(e^{x} = u\) and \(dx = \frac{du}{u}\), so the integral becomes \(\int u \sec^{3}(u) \cdot \frac{du}{u} = \int \sec^{3}(u) \, du\).

Now, focus on evaluating \(\int \sec^{3}(u) \, du\). Recall that integrals of odd powers of secant can be handled by splitting \(\sec^{3}(u)\) as \(\sec(u) \cdot \sec^{2}(u)\) and using the identity \(\sec^{2}(u) = 1 + \tan^{2}(u)\).

Use the substitution \(t = \tan(u)\), so that \(dt = \sec^{2}(u) du\). Express the integral in terms of \(t\) and integrate accordingly, then substitute back to \(u\) and finally back to \(x\) to express the answer in terms of the original variable.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. It is useful when the integral contains a composite function, allowing you to rewrite the integral in terms of a new variable, making it easier to solve.

Trigonometric Identities for Secant

Understanding identities involving secant, such as expressing sec³(x) in terms of sec(x) and tan(x), helps in breaking down complex trigonometric integrals. These identities facilitate rewriting the integral into simpler parts that are easier to integrate.

Integration of Powers of Secant

Integrating powers of secant functions often requires specific techniques, like using reduction formulas or rewriting powers in terms of sec and tan. Familiarity with these methods is essential to evaluate integrals involving sec³(x) or higher powers.

Evaluate the integrals in Exercises 33–52.∫ eˣ sec³(eˣ) dx