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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 5.3.59
Chapter 5, Problem 5.3.59

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


∫₁² (z² + 4) / z dz

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Step 1: Simplify the integrand. The given integral is ∫₁² (z² + 4) / z dz. Split the fraction into two terms: (z² / z) + (4 / z). This simplifies to ∫₁² (z + 4/z) dz.
Step 2: Break the integral into two separate integrals. Using the linearity property of integrals, rewrite the expression as ∫₁² z dz + ∫₁² (4/z) dz.
Step 3: Compute the antiderivative of each term. For the first term, ∫ z dz, the antiderivative is (1/2)z². For the second term, ∫ (4/z) dz, the antiderivative is 4ln|z|.
Step 4: Apply the Fundamental Theorem of Calculus. Evaluate the antiderivatives at the upper limit (z = 2) and subtract the evaluation at the lower limit (z = 1). For (1/2)z², compute [(1/2)(2²) - (1/2)(1²)]. For 4ln|z|, compute [4ln(2) - 4ln(1)].
Step 5: Combine the results from both terms. Add the results from the evaluations of (1/2)z² and 4ln|z| to obtain the final value of the definite integral.

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Key Concepts

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

Definite Integrals

Definite integrals represent the signed area under a curve between two specified limits. They are calculated using the integral symbol with lower and upper bounds, indicating the interval over which the function is evaluated. The result of a definite integral is a numerical value that reflects the accumulation of quantities, such as area, over that interval.
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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links the concept of differentiation with integration, providing a method to evaluate definite integrals. It states that if a function is continuous on an interval, the integral of its derivative over that interval equals the difference in the values of the original function at the endpoints. This theorem allows for the computation of definite integrals by finding an antiderivative of the integrand.
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Antiderivatives

An antiderivative of a function is another function whose derivative yields the original function. Finding an antiderivative is essential for evaluating definite integrals using the Fundamental Theorem of Calculus. For example, if F(z) is an antiderivative of f(z), then the definite integral from a to b can be computed as F(b) - F(a), providing the net area under the curve of f(z) between those limits.
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Related Practice
Textbook Question

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  ∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)

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Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    

                                                                                                                                                                    

  ∫ d𝓍 / [√1 + √(1 + 𝓍)] (Hint: Begin with u = √(1 + 𝓍 .)  

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Textbook Question

Determine the intervals on which the function g(𝓍) = ∫ₓ⁰ t / (t² + 1) dt  is concave up or concave down.

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Textbook Question

Evaluate


lim [ ∫₂ˣ √(t² + t + 3dt) ] / (𝓍² ―4)

𝓍→2

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Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus


∫₁² 3/t dt

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Textbook Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.


 ∫₀⁴ ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5      if 𝓍 ≤ 2                                                                                                                                                                                     

                      3𝓍 ― 1  if 𝓍 > 2

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