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 8.6.68
Evaluate the following integrals.
∫ eˣ/(e²ˣ + 2eˣ + 17) dx
Guida verificata passo dopo passo1
Step 1: Observe the integral ∫ eˣ/(e²ˣ + 2eˣ + 17) dx. Notice that the denominator is a quadratic expression in terms of eˣ. Rewrite the denominator as (eˣ)² + 2(eˣ) + 17 to make the structure clearer.
Step 2: Perform a substitution to simplify the integral. Let u = eˣ, which implies that du = eˣ dx. This substitution transforms the integral into ∫ u/(u² + 2u + 17) du.
Step 3: Analyze the new integral ∫ u/(u² + 2u + 17) du. Factorize or complete the square for the quadratic expression in the denominator. Rewrite u² + 2u + 17 as (u + 1)² + 16.
Step 4: Rewrite the integral using the completed square form: ∫ u/((u + 1)² + 16) du. Consider splitting the numerator u into two terms: (u + 1) - 1, so the integral becomes ∫ [(u + 1)/((u + 1)² + 16) - 1/((u + 1)² + 16)] du.
Step 5: Break the integral into two parts: ∫ (u + 1)/((u + 1)² + 16) du and ∫ -1/((u + 1)² + 16) du. Use standard integration techniques for these forms, such as recognizing the derivative of the denominator for the first term and using the arctangent formula for the second term.
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Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is crucial for evaluating complex integrals, such as the one presented, where a rational function is involved.
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Percorso guidato
06:18
Integration by Parts for Definite Integrals
Exponential Functions
Exponential functions are mathematical functions of the form f(x) = e^x, where e is Euler's number (approximately 2.718). These functions have unique properties, such as their derivatives being equal to the function itself. Recognizing the behavior of exponential functions is essential when integrating expressions that involve e^x.
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Rational Functions
A rational function is a ratio of two polynomials. In the integral provided, the denominator is a polynomial that can be factored or simplified to aid in integration. Understanding how to manipulate and simplify rational functions is key to solving integrals that involve them, especially when combined with exponential terms.
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Pratica correlata
Domanda del libro di testo
79. Tabular integration extended Refer to Exercise 77.
a. The following table shows the method of tabular integration applied to
∫ eˣ cos x dx.
Use the table to express ∫ eˣ cos x dx in terms of the sum of functions and an indefinite integral.
b. Solve the equation in part (a) for ∫ eʳ cos z dz.
c. Evaluate ∫ e⁻ᶻ sin 3z dz by applying the idea from parts (a) and (b).
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Domanda del libro di testo
5–16. Set up the appropriate form of the partial fraction decomposition for the following expressions. Do not find the values of the unknown constants.
6. (4x + 1)/(4x² - 1)
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Domanda del libro di testo
85. Another form of ∫ sec x dx
a. Verify the identity:
sec x = cos x / (1 - sin² x)
b. Use the identity in part (a) to verify that:
∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C
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Domanda del libro di testo
6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.
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Domanda del libro di testo
69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved about the y-axis?
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