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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.5.119
Chapter 5, Problem 5.5.119

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    
                                                                                                                                                                    
  ∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)

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Step 1: Begin by making the substitution u = cos(θ). This implies that du = -sin(θ)dθ. Rewrite the integral in terms of u, noting that when θ = 0, u = cos(0) = 1, and when θ = π/2, u = cos(π/2) = 0.
Step 2: Substitute u = cos(θ) and du = -sin(θ)dθ into the integral. The integral becomes ∫₁⁰ (-u / √(u² + 16)) du. The negative sign can be used to reverse the limits of integration, changing the integral to ∫₀¹ (u / √(u² + 16)) du.
Step 3: To simplify further, consider a second substitution. Let v = u² + 16, which implies dv = 2u du. Rewrite the integral in terms of v, noting that when u = 0, v = 16, and when u = 1, v = 17.
Step 4: Substitute v = u² + 16 and dv = 2u du into the integral. The integral becomes (1/2) ∫₁⁷ (1 / √v) dv. The factor of 1/2 comes from the substitution dv = 2u du.
Step 5: Evaluate the integral ∫₁⁷ (1 / √v) dv using the formula for the integral of 1/√v, which is 2√v. Substitute the limits of integration (v = 16 and v = 17) into the result to complete the solution.

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Key Concepts

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

Substitution in Integration

Substitution is a technique used in integration to simplify the integral by changing the variable of integration. By substituting a new variable, often denoted as 'u', for a function of the original variable, the integral can become easier to evaluate. This method is particularly useful when dealing with composite functions or when the integrand contains complicated expressions.
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Substitution With an Extra Variable

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. These identities, such as sin²θ + cos²θ = 1, can be used to simplify integrals involving trigonometric functions. Understanding these identities is crucial for manipulating expressions and making substitutions in integrals that contain trigonometric terms.
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Verifying Trig Equations as Identities

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as ∫[a,b] f(x) dx, where 'a' and 'b' are the limits of integration. Evaluating definite integrals often involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus, which connects differentiation and integration.
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Related Practice
Textbook Question

{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.

The midpoint Riemann sum for f(x) = x³ on [3,11] with n = 32.

Textbook Question

Variations on the substitution method Evaluate the following integrals.                                                                                                        

                                                                                                                                                                    

 ∫ 𝓍/(√𝓍―4) d𝓍

Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 ∫ [ 1/(10𝓍―3) d𝓍

Textbook Question

Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2 and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.

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Textbook Question

Explain why ∫ₐᵇ ƒ ′(𝓍) d𝓍 = ƒ(b) ― ƒ(a)

Textbook Question

Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = t² + 3t. Find the average velocity of the object over this time interval.