Textbook Question
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
∫₁⁸ 8𝓍¹/³ d𝓍
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8. Definite Integrals
Fundamental Theorem of Calculus
1:44 minutesProblem 5.3.47
Textbook Question
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
∫₁² 3/t dt
Verified step by step guidance1
Step 1: Identify the integral to be evaluated. The problem asks us to compute the definite integral ∫₁² (3/t) dt using the Fundamental Theorem of Calculus.
Step 2: Recall the Fundamental Theorem of Calculus, which states that if F(x) is an antiderivative of f(x), then ∫ₐᵇ f(x) dx = F(b) - F(a).
Step 3: Find the antiderivative of the integrand 3/t. The antiderivative of 1/t is ln|t|, so the antiderivative of 3/t is 3 * ln|t|.
Step 4: Apply the Fundamental Theorem of Calculus. Substitute the limits of integration into the antiderivative: F(2) - F(1), where F(t) = 3 * ln|t|.
Step 5: Simplify the expression by evaluating 3 * ln|2| - 3 * ln|1|. Note that ln|1| equals 0, so the result simplifies further.
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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 quantifies this area, which can be interpreted in various contexts, such as physics and economics.
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Definition of the Definite Integral
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and 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 us to compute definite integrals by finding an antiderivative of the integrand.
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Antiderivatives
An antiderivative of a function is another function whose derivative is the original function. Finding an antiderivative is essential for evaluating definite integrals using the Fundamental Theorem of Calculus. For example, if we need to integrate a function like 3/t, we seek a function whose derivative gives us 3/t, which in this case is 3 ln|t|. Evaluating the definite integral then involves substituting the limits into this antiderivative.
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