Domanda del libro di testo
7–84. Evaluate the following integrals.
45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.6.71
Evaluate the following integrals.
∫ (1 - cosx)/(1 + cosx) dx
Guida verificata passo dopo passo1
Step 1: Recognize that the integral involves trigonometric functions. To simplify the integrand, consider using trigonometric identities. Specifically, recall that 1 + cos(x) can be expressed as 2cos²(x/2) and 1 - cos(x) can be expressed as 2sin²(x/2). This substitution will help simplify the fraction.
Step 2: Rewrite the integrand using the trigonometric identities: ∫ (1 - cos(x))/(1 + cos(x)) dx = ∫ (2sin²(x/2))/(2cos²(x/2)) dx. Simplify the fraction to get ∫ tan²(x/2) dx.
Step 3: Use the identity tan²(u) = sec²(u) - 1 to rewrite the integrand. This gives ∫ tan²(x/2) dx = ∫ (sec²(x/2) - 1) dx.
Step 4: Split the integral into two parts: ∫ sec²(x/2) dx - ∫ 1 dx. The first term, ∫ sec²(x/2) dx, integrates to (2tan(x/2)) because the derivative of tan(x/2) is sec²(x/2) multiplied by the chain rule factor of 1/2. The second term, ∫ 1 dx, integrates to x.
Step 5: Combine the results from the two integrals and include the constant of integration, C. The final expression will be in terms of x and tan(x/2).
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Integration
Integration is a fundamental concept in calculus that involves finding the accumulated area under a curve represented by a function. It is the reverse process of differentiation and can be used to calculate definite and indefinite integrals. Understanding integration techniques, such as substitution and integration by parts, is essential for solving integral problems.
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Integration by Parts for Definite Integrals
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are crucial for simplifying expressions and solving integrals involving trigonometric functions. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas, which can help in transforming the integrand into a more manageable form.
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Substitution Method
The substitution method is a technique used in integration to simplify the integrand by changing variables. This method involves substituting a part of the integral with a new variable, which can make the integral easier to evaluate. It is particularly useful when dealing with composite functions or when the integrand can be expressed in a simpler form through a suitable substitution.
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