Question

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $$ g(x)=\int^{x} \ln \left(1+t^{2}\right) d t $$

Name: Problem 10, Chapter 5: Calculus: Early Transcendentals, Metric Edition
Uploaded: 2022-09-18T17:39:50-08:00
Channel: Suzanne W.
Description: Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. g(x)=∫^x ln(1+t^2) d t

   Use Part 1 of the Fundamental Theorem of Calculus to find
the derivative of the function.
$$
g(x)=\int^{x} \ln \left(1+t^{2}\right) d t
$$

Calculus: Early Transcendentals, Metric Edition

James Stewart,… 9th Edition

Chapter 5, Problem 10

Instant Answer

Solved by Expert Suzanne W.

09/18/2022

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In this case, the function is $f(t) = \ln(1+t^2)$. Show more…

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Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $$ g(x)=\int^{x} \ln \left(1+t^{2}\right) d t $$

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Key Concepts

Fundamental Theorem of Calculus (Part 1)

This theorem establishes that if a function is defined as an integral with a variable upper limit, then its derivative is simply the integrand evaluated at that upper limit. This relationship links the operations of integration and differentiation, showing that they are inverse processes under certain conditions.

Differentiation of Integral Functions

This concept involves finding the derivative of a function defined by an integral with a variable limit. When the upper limit of the integral is the variable of differentiation, the derivative is obtained by directly substituting the variable into the integrand, assuming the conditions of the fundamental theorem are met.

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