How do you find the derivative of integral of #x^2*sin(t^2)dt# from 0 to x?

Answer 1

I can answer if you meant to ask:

How do you find the derivative of integral of #t^2*sin(t^2)dt# from 0 to x?
This is a problem for the Fundamental Theorem of Calculus, Part 1, which says: If #f(x)# is defined by:
#f(x) = int_a^x g(t) dt# for #x# in some #[a,b]#, then
#g'(x) = f(x)#.

The job (one job) of this theorem is to tell us that this is the easiest question on the examination:

Find the derivative #int_0^x t^2*sin(t^2)dt#
The answer is: #x^2sin(x^2)#

Now move on to the next question.

What is the derivative of #int_3^x (t-sint)^3/(t^2+4t+4)dt#?
It is #(x-sinx)^3/(x^2+4x+4)#.
The only thing that can go wrong is you forget to change the #t#'s to #x#'s.
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Answer 2

To find the derivative of the integral of ( \int_{0}^{x} x^2 \cdot \sin(t^2) , dt ) with respect to ( x ), you can use the Fundamental Theorem of Calculus along with the chain rule.

The derivative of the integral is ( x^2 \cdot \sin(x^2)^2 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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