Derivatives of integrals Simplify the following expressions.

d/d𝓍 ∫₃ˣ (t² + t + 1) dt

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Recognize that the problem involves the Fundamental Theorem of Calculus, which states that if F(x) = ∫ₐˣ f(t) dt, then dF/dx = f(x).

Identify the integral bounds: the lower bound is a constant (3), and the upper bound is the variable (x). This means the derivative will be evaluated at the upper bound.

Apply the Fundamental Theorem of Calculus: d/d𝓍 ∫₃ˣ (t² + t + 1) dt = (t² + t + 1) evaluated at t = x.

Substitute t = x into the integrand: (x² + x + 1).

Conclude that the derivative simplifies to x² + x + 1, as the lower bound being constant does not contribute to the derivative.

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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, then the integral of f from a to b can be computed using F(b) - F(a). This theorem is crucial for simplifying expressions involving derivatives of integrals, as it allows us to evaluate the derivative of an integral directly.

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 managing the relationship between the outer function (the derivative) and the inner function (the integral), ensuring that we account for changes in the variable of integration.

Definite Integral

A definite integral represents the accumulation of a function's values over a specific interval, providing a numerical result. In the context of the given expression, understanding how to evaluate the definite integral from 3 to x is essential for applying the Fundamental Theorem of Calculus and simplifying the expression correctly.

Derivatives of integrals Simplify the following expressions.d/d𝓍 ∫₃ˣ (t² + t + 1) dt