Chapter . 5.4.33
Problem Statement
Calculate the derivative.
Step 1
Let %3D%40INTEGRAL%7B%40RT%7Bx%7D%3Bx%40SUP%7B2%7D%7Dtan%24a%40I%7Bt%7D%40T%7B%7Ddt&doborder=false&bottom=0)
.
In order to use the Fundamental Theorem of Calculus (Part 2), G(x) needs to consist of integrals of the form %7Df(t)dt&doborder=false&bottom=0)
, where a is any constant.
Question Sequence
Question
Use properties of integrals to write G(x) as a sum or difference of integrals in which the lower limit or each integral is a = 0.
%3D%40INTEGRAL%7B%40RT%7Bx%7D%3Bx%40SUP%7B2%7D%7Dtan%24a%40I%7Bt%7D%40T%7B%7Ddt%3D&doborder=false&bottom=0)
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
Correct.
That's not right. Check your work.
Incorrect.
Step 2
The Chain Rule for differentiating a function of the form %3D%40INTEGRAL%7Ba%3Bu%7Dg(t)dt&doborder=false&bottom=0)
, where &doborder=false&bottom=0)
is
%3D%40DIV%7Bd%3Bdx%7D%40INTEGRAL%7Ba%3Bu%7Dg(t)dt%3Dg(u)%40DIV%7Bdu%3Bdx%7D&doborder=false&bottom=0)
.
Question Sequence
Question
Calculate the derivative, G'(x).
%3D%40DIV%7Bd%3Bdx%7D%40INTEGRAL%7B%40RT%7Bx%7D%3Bx%40SUP%7B2%7D%7Dtan%24a%40I%7Bt%7D%40T%7B%7Ddt&doborder=false&bottom=0)
-%40DIV%7Bd%3Bdx%7D(%40INTEGRAL%7B0%3B%40RT%7Bx%7D%7Dtan%24a%40I%7Bt%7D%40T%7B%7Ddt)%3D&doborder=false&bottom=0)
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
Incorrect. Do you see where you went wrong?
Excellent work.