`h(x) = int_1^(sqrt(x))(z^2/(z^4 + 1))dz` Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.

Textbook Question

Chapter 5, 5.3 - Problem 14 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Hello!

Part 1 of the Fundamental Theorem of Calculus states that for a continuous function `f`
`F'_a(x)=f(x),` where `F_a(x)=int_a^xf(t)dt.`

 

Here `f(t)=t^2/(t^4+1)` and `h(x)=F_1(sqrt(x))`

(x>=0).

 

Therefore

`h'(x)=d/(dx)(h(x))=d/(dx)(F_1(sqrt(x)))=F'_1(sqrt(x))*1/(2sqrt(x))=f(sqrt(x))*1/(2sqrt(x))=`

`=x/(x^2+1)*1/(2sqrt(x))=(sqrt(x))/(2(x^2+1)).`

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