use part 1 of the fundamental theorem of calculus to find the derivative of the function:
integral[sqrt(5+r^3)]dr from 2 to x^2
I'm stuck on this problem, can someone explain the steps to do the problem correctly? thanks!
integral[sqrt(5+r^3)]dr from 2 to x^2
I'm stuck on this problem, can someone explain the steps to do the problem correctly? thanks!
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Let F(x) = integral [√(5+r^3)]dr from 2 to x
The fundamental theorem says that F'(x) = √(5 + x^3)
Let G(x) = integral [√(5+r^3)]dr from 2 to x^2
Then G(x) = F(x^2)
By the chain rule
G'(x) = F'(x^2) * (x^2)' = √(5 + (x^2)^3) * 2x = 2x√(5 + x^6)
The fundamental theorem says that F'(x) = √(5 + x^3)
Let G(x) = integral [√(5+r^3)]dr from 2 to x^2
Then G(x) = F(x^2)
By the chain rule
G'(x) = F'(x^2) * (x^2)' = √(5 + (x^2)^3) * 2x = 2x√(5 + x^6)
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The fundamental theorem says (if I remember well) that if you differentiate an integral, you get back the original function.
Therefore, the derivative of "Integral of XYZ" is simply XYZ
because when you integrate XYZ, you get "Integral of XYZ"
Derivative will be
√(5 + r^3)
at r=2, the derivative will be √(5 + 8) = √(13)
at r=x^2, the derivative will be √(5 + x^6)
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"the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation"
In other words: differentiating removes the integral.
"the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives."
Normally, when one is given a definite integral (an integral, followed with borders, such as "from 2 to x^2"), they are asked to evalutate the integral, not to find the derivative.
However, since your questions states "use part 1 of the fundamental theorem", I assume that you want to find the derivative of the integral (meaning: remove the integral sign and the "dr"; keep what is left).
Therefore, the derivative of "Integral of XYZ" is simply XYZ
because when you integrate XYZ, you get "Integral of XYZ"
Derivative will be
√(5 + r^3)
at r=2, the derivative will be √(5 + 8) = √(13)
at r=x^2, the derivative will be √(5 + x^6)
---
"the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation"
In other words: differentiating removes the integral.
"the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives."
Normally, when one is given a definite integral (an integral, followed with borders, such as "from 2 to x^2"), they are asked to evalutate the integral, not to find the derivative.
However, since your questions states "use part 1 of the fundamental theorem", I assume that you want to find the derivative of the integral (meaning: remove the integral sign and the "dr"; keep what is left).