Textbook Question
2β74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
18. β« (from 0 to β2) (x + 1)/(3xΒ² + 6) dx
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8. Definite Integrals
Substitution
5:00 minutesProblem 5.R.104
Textbook Question
Change of variables Use the change of variables uΒ³ = πΒ² β 1 to evaluate the integral β«βΒ³ πβ(πΒ²β1) dπ .
Verified step by step guidance1
Step 1: Identify the substitution. Let uΒ³ = πΒ² - 1. Differentiate both sides with respect to π to find the relationship between du and dπ. Differentiating gives 3uΒ² du = 2π dπ.
Step 2: Solve for dπ in terms of u and du. Rearrange the equation to get dπ = (3uΒ²)/(2π) du.
Step 3: Rewrite π in terms of u using the substitution uΒ³ = πΒ² - 1. Solving for π gives π = β(uΒ³ + 1).
Step 4: Change the limits of integration. When π = 1, uΒ³ = 1Β² - 1 = 0, so u = 0. When π = 3, uΒ³ = 3Β² - 1 = 8, so u = 2.
Step 5: Substitute everything into the integral. Replace π, dπ, and the integrand with their expressions in terms of u. The integral becomes β«βΒ² β(uΒ³ + 1) β(uΒ³) * (3uΒ²)/(2β(uΒ³ + 1)) du. Simplify the integrand and proceed to evaluate the integral.
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5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Change of Variables
The change of variables technique in calculus allows us to simplify integrals by substituting a new variable for the original variable. This method can transform a complex integral into a more manageable form, making it easier to evaluate. The substitution must be accompanied by the appropriate adjustment of the differential, ensuring that the limits of integration and the integrand are correctly modified.
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Definite Integrals
A definite integral represents the signed area under a curve between two specified limits. It is denoted as β«_a^b f(x) dx, where 'a' and 'b' are the lower and upper limits, respectively. Evaluating a definite integral involves finding the antiderivative of the function and then applying the Fundamental Theorem of Calculus to compute the difference between the values at the limits.
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Integration by Substitution
Integration by substitution is a method used to simplify the process of integration by changing the variable of integration. This technique often involves identifying a part of the integrand that can be replaced with a single variable, which simplifies the integral. The derivative of the substituted variable must also be accounted for, ensuring that the integral remains equivalent to the original.
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