Evaluate the integrals in Exercises 37–44.
∫ eᵗ √[tan²(eᵗ) + 1] dt

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Recognize that the integral is of the form \(\int e^{t} \sqrt{\tan^{2}(e^{t}) + 1} \, dt\). Notice that inside the square root, we have \(\tan^{2}(e^{t}) + 1\), which can be simplified using a trigonometric identity.

Recall the Pythagorean identity: \(1 + \tan^{2}(x) = \sec^{2}(x)\). Applying this, rewrite the integrand as \(e^{t} \sqrt{\sec^{2}(e^{t})}\).

Since \(\sqrt{\sec^{2}(e^{t})} = |\sec(e^{t})|\), and assuming the domain where \(\sec(e^{t})\) is positive, the integrand simplifies to \(e^{t} \sec(e^{t})\).

Use substitution to simplify the integral: let \(u = e^{t}\). Then, \(du = e^{t} dt\), which means \(du = e^{t} dt\) or equivalently \(e^{t} dt = du\). Substitute these into the integral to get \(\int \sec(u) \, du\).

Now, the integral reduces to \(\int \sec(u) \, du\), which is a standard integral. You can proceed by recalling the formula for \(\int \sec(u) \, du\) and then substitute back \(u = e^{t}\) to express the answer in terms of \(t\).

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주요 개념

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

Integration of Exponential Functions

Understanding how to integrate functions involving exponential terms like e^t is essential. This includes recognizing when substitution can simplify the integral, especially when the exponent appears inside another function.

Trigonometric Identities

Familiarity with trigonometric identities, such as 1 + tan²(x) = sec²(x), helps simplify expressions under the integral. Applying these identities can transform complex radicals into more manageable forms.

Substitution Method in Integration

The substitution method involves changing variables to simplify the integral. Identifying an inner function whose derivative appears elsewhere in the integrand allows rewriting the integral in a simpler form for direct integration.

Evaluate the integrals in Exercises 37–44.∫ eᵗ √[tan²(eᵗ) + 1] dt