Differentiate F(x)=∫₀ˣ(t²+1)dt.
Show a complete solution and verify with a second representation.
Unit 10 · Core Pathway
Fundamental Theorem & Antiderivatives
How does the Fundamental Theorem connect instantaneous rate with accumulated change?
Khaemenes Academy · Calculus I
Core Practice
Fundamental Theorem & Antiderivatives · Show reasoning, notation, intervals, units, and corrections.
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Differentiate H(x)=∫ₓ³ e^t dt.
Show a complete solution and verify with a second representation.
Why use a dummy variable inside an accumulation integral?
Show a complete solution and verify with a second representation.
Evaluate ∫₀² 3x² dx using FTC.
Show a complete solution and verify with a second representation.
Why use a dummy variable inside an accumulation integral?
Show a complete solution and verify with a second representation.
Evaluate ∫₀² 3x² dx using FTC.
Show a complete solution and verify with a second representation.
What connects derivatives and integrals?
Show a complete solution and verify with a second representation.
Find ∫(3x²−2)dx.
Show a complete solution and verify with a second representation.
Find ∫e^x dx.
Show a complete solution and verify with a second representation.
Find ∫(3x²−2)dx.
Show a complete solution and verify with a second representation.
Why include +C?
Show a complete solution and verify with a second representation.
Evaluate ∫2x(x²+1)³ dx.
Show a complete solution and verify with a second representation.
Evaluate ∫cos(3x)dx.
Show a complete solution and verify with a second representation.
Evaluate ∫2x(x²+1)³ dx.
Show a complete solution and verify with a second representation.
Choose u for ∫cos(3x)dx.
Show a complete solution and verify with a second representation.
How is substitution verified?
Show a complete solution and verify with a second representation.