Textbook Question
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
∫π/₄^³π/⁴ (cot² 𝓍 + 1) d𝓍
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8. Definite Integrals
Fundamental Theorem of Calculus
2:39 minutesProblem 5.3.82
Textbook Question
Derivatives of integrals Simplify the following expressions.
d/dy ∫¹⁰ᵧ³ √(𝓍⁶ + 1) d𝓍
Verified step by step guidance1
Step 1: Recognize that the problem involves the Fundamental Theorem of Calculus, which states that if F(x) is the antiderivative of f(x), then d/dx ∫ₐˣ f(t) dt = f(x).
Step 2: Observe that the integral has variable limits of integration (y³ as the lower limit and 10 as the upper limit). This requires the use of Leibniz's rule for differentiation under the integral sign.
Step 3: Apply Leibniz's rule: d/dy ∫ₐᵇ f(x) dx = f(b) * (db/dy) - f(a) * (da/dy), where a and b are functions of y. Here, a = y³ and b = 10.
Step 4: Substitute the limits into the formula. The upper limit (10) is constant, so db/dy = 0. For the lower limit (y³), da/dy = d(y³)/dy = 3y². The integrand is √(x⁶ + 1), so evaluate it at x = y³.
Step 5: Simplify the expression using the formula: d/dy ∫¹⁰ᵧ³ √(x⁶ + 1) dx = -√((y³)⁶ + 1) * 3y². The negative sign comes from the lower limit contribution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then the integral of f from a to b can be computed using F. It also establishes that the derivative of the integral of a function is the original function itself, which is crucial for simplifying expressions involving derivatives of integrals.
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Chain Rule
The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. When differentiating an integral with variable limits, the Chain Rule helps in applying the derivative to the upper limit while also considering the derivative of the limit itself. This is essential for correctly simplifying expressions where the limits of integration are functions of a variable.
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Definite Integral
A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. In the context of the given expression, understanding how to evaluate definite integrals and their properties is necessary for simplifying the expression involving the integral of √(𝓍⁶ + 1) from 1 to 3y. This knowledge is key to applying the Fundamental Theorem of Calculus effectively.
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