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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.3.57
Chapter 8, Problem 8.3.57

9–61. Trigonometric integrals Evaluate the following integrals.
57. ∫ from 0 to π of (1 - cos2x)³ᐟ² dx

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Step 1: Recognize that the integral involves a trigonometric expression raised to a fractional power. To simplify, use the trigonometric identity: cos(2x) = 1 - 2sin²(x). Rewrite 1 - cos(2x) as 2sin²(x).
Step 2: Substitute 1 - cos(2x) with 2sin²(x) in the integral. The integral becomes: ∫ from 0 to π of (2sin²(x))³ᐟ² dx.
Step 3: Simplify the expression (2sin²(x))³ᐟ². This can be rewritten as (2³ᐟ²)(sin²(x))³ᐟ², where 2³ᐟ² is a constant factor that can be factored out of the integral.
Step 4: Rewrite the integral as 2³ᐟ² ∫ from 0 to π of (sin(x))² dx. To evaluate this integral, use the power-reduction formula for sine: sin²(x) = (1 - cos(2x))/2. Substitute this into the integral.
Step 5: After substitution, the integral becomes 2³ᐟ² ∫ from 0 to π of (1 - cos(2x))/2 dx. Split the integral into two parts: (2³ᐟ²/2) ∫ from 0 to π of 1 dx and -(2³ᐟ²/2) ∫ from 0 to π of cos(2x) dx. Evaluate each part separately using basic integration rules.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus, particularly in integrals involving periodic functions. Understanding their properties, such as periodicity and symmetry, is crucial for evaluating integrals that include these functions. In this case, the integral involves the cosine function, which can be manipulated using trigonometric identities.
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Introduction to Trigonometric Functions

Integration Techniques

Integration techniques, including substitution and integration by parts, are essential for solving complex integrals. In this problem, recognizing the structure of the integrand (1 - cos(2x))^3/2 may suggest a substitution or a trigonometric identity to simplify the integral. Mastery of these techniques allows for the effective evaluation of integrals that may initially appear daunting.
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Definite Integrals

Definite integrals calculate the area under a curve between specified limits, in this case, from 0 to π. Understanding the properties of definite integrals, such as the Fundamental Theorem of Calculus, is vital for evaluating the integral accurately. This theorem connects differentiation and integration, allowing for the evaluation of the integral by finding an antiderivative and applying the limits.
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Related Practice
Textbook Question

7–84. Evaluate the following integrals.

45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

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Textbook Question

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).

Textbook Question

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

Textbook Question

Evaluate the following integrals.

∫ (1 - cosx)/(1 + cosx) dx

Textbook Question

6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

Textbook Question

23-64. Integration Evaluate the following integrals.

60.∫ 1/[(y² + 1)(y² + 2)] dy

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