Differentiate F(x)=∫₀ˣ(t²+1)dt.
Generalize, analyze an edge case, or defend the result without relying on technology.
Unit 10 · Extended Pathway
Fundamental Theorem & Antiderivatives
How does the Fundamental Theorem connect instantaneous rate with accumulated change?
Khaemenes Academy · Calculus I
Extended Practice
Fundamental Theorem & Antiderivatives · Show reasoning, notation, intervals, units, and corrections.
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What connects derivatives and integrals?
Generalize, analyze an edge case, or defend the result without relying on technology.
Differentiate G(x)=∫₁^{x²} cos t dt.
Generalize, analyze an edge case, or defend the result without relying on technology.
Why use a dummy variable inside an accumulation integral?
Generalize, analyze an edge case, or defend the result without relying on technology.
Differentiate H(x)=∫ₓ³ e^t dt.
Generalize, analyze an edge case, or defend the result without relying on technology.
Find ∫(3x²−2)dx.
Generalize, analyze an edge case, or defend the result without relying on technology.
Find ∫e^x dx.
Generalize, analyze an edge case, or defend the result without relying on technology.
Find F if F'=2x and F(1)=5.
Generalize, analyze an edge case, or defend the result without relying on technology.
Evaluate ∫2x(x²+1)³ dx.
Generalize, analyze an edge case, or defend the result without relying on technology.
Why change bounds in a definite substitution?
Generalize, analyze an edge case, or defend the result without relying on technology.
Evaluate ∫₀¹ 2x e^(x²) dx.
Generalize, analyze an edge case, or defend the result without relying on technology.
How is substitution verified?
Generalize, analyze an edge case, or defend the result without relying on technology.