Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₋₁⁰ e^x / (e^x + e^(−x)) dx
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.PE.44
Evaluate the integrals in Exercises 37–44.
∫ eᵗ √[tan²(eᵗ) + 1] dt
Guida verificata passo dopo passo1
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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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.
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Integrals of General Exponential Functions
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.
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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.
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