Differentiate an accumulated integral
When the integrand is continuous on the relevant interval, differentiating an integral with a variable upper endpoint evaluates the integrand at that endpoint.
Calculus / English
Ex 7: Second Fundamental Theorem of Calculus with Chain Rule
Mathispower4u · YouTube · 4:16
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Combine the fundamental theorem of calculus with the chain rule when an integral has a square-root upper limit.
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0:00Differentiate an accumulated integral0:50Differentiate the endpoint too3:45Simplify after substitutionLearning script
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When the integrand is continuous on the relevant interval, differentiating an integral with a variable upper endpoint evaluates the integrand at that endpoint.
The endpoint is u=√t, so for its derivative is 1/(2√t). Multiply this chain-rule factor by the integrand evaluated at u.
For the lesson’s integrand sin(x⁵)/x⁴ and constant lower endpoint 2, the derivative at is . The reciprocal endpoint-derivative factor is essential.
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Differentiate the endpoint too
The endpoint is u=√t, so for its derivative is 1/(2√t). Multiply this chain-rule factor by the integrand evaluated at u.
Simplify after substitution
For the lesson’s integrand sin(x⁵)/x⁴ and constant lower endpoint 2, the derivative at is . The reciprocal endpoint-derivative factor is essential.
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- Chain rule ApplicationAt 0:00
Why this connection?
When the integrand is continuous on the relevant interval, differentiating an integral with a variable upper endpoint evaluates the integrand at that endpoint. The endpoint is u=√t, so for its derivative is 1/(2√t). Multiply this chain-rule factor by the integrand evaluated at u. For the lesson’s integrand sin(x⁵)/x⁴ and constant lower endpoint 2, the derivative at is . The reciprocal endpoint-derivative factor is essential.
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