Textbook Question
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus
∫₁⁴ (𝓍 ― 2)/√𝓍 d𝓍
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8. Definite Integrals
Fundamental Theorem of Calculus
2:27 minutesProblem 5.3.81
Textbook Question
Derivatives of integrals Simplify the following expressions.
d/dz ∫¹⁰ₛᵢₙ ₂ dt /(t⁴ + 1)
Verified step by step guidance1
Step 1: Recognize that the problem involves the derivative of an integral. This is a classic application of the Fundamental Theorem of Calculus, which states that if F(x) = ∫[a(x), b(x)] f(t) dt, then dF/dx = f(b(x)) * b'(x) - f(a(x)) * a'(x).
Step 2: Identify the bounds of the integral. The upper bound is 10, which is constant, and the lower bound is sin(z), which is a function of z. This means the derivative will involve the lower bound's derivative.
Step 3: Apply the Fundamental Theorem of Calculus. Since the upper bound is constant, its derivative contributes nothing. For the lower bound, substitute t = sin(z) into the integrand f(t) = 1/(t⁴ + 1), and multiply by the derivative of sin(z), which is cos(z).
Step 4: Write the expression for the derivative. The result is -[1/(sin(z)⁴ + 1)] * cos(z). The negative sign comes from the fact that the lower bound contributes negatively in the Fundamental Theorem of Calculus.
Step 5: Simplify the expression if needed. The derivative is now expressed in terms of z, and no further simplification is required unless specified.
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2mKey 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 as F(b) - F(a). This theorem also implies that the derivative of an integral function is the integrand evaluated at the upper limit of integration, which is crucial for simplifying expressions involving derivatives of integrals.
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Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as ∫_a^b f(x) dx and provides a numerical value that reflects the accumulation of quantities, such as area or total change, over that interval. Understanding how to evaluate definite integrals is essential for applying the Fundamental Theorem of Calculus.
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Chain Rule
The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative dy/dx can be found by multiplying the derivative of the outer function f with respect to g by the derivative of the inner function g with respect to x. This rule is particularly useful when dealing with integrals that have variable limits, as seen in the given expression.
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