# How do you use the second fundamental theorem of Calculus to find the derivative of given #int (t^2 +3t+2)dt# from #[-3, x]#?

The second Fundamental Theorem of Calculus enables us to find the function defined by

Note We get the same answer much more quickly by using part 1 of the fundamental theorem.

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To use the second fundamental theorem of calculus to find the derivative of ( \int_{-3}^{x} (t^2 + 3t + 2) dt ), you first need to find the antiderivative of the function ( t^2 + 3t + 2 ). Then, you evaluate this antiderivative at the upper limit ( x ) and subtract the value of the antiderivative evaluated at the lower limit (-3).

So, let ( F(t) ) be the antiderivative of ( t^2 + 3t + 2 ). Then, by the second fundamental theorem of calculus, the derivative of ( \int_{-3}^{x} (t^2 + 3t + 2) dt ) with respect to ( x ) is given by:

[ \frac{d}{dx} \left[ \int_{-3}^{x} (t^2 + 3t + 2) dt \right] = F(x) - F(-3) ]

Where ( F(x) ) is the antiderivative of ( t^2 + 3t + 2 ). To find ( F(x) ), integrate ( t^2 + 3t + 2 ) with respect to ( t ). After finding ( F(x) ), evaluate it at ( x ) and subtract the value of ( F(-3) ). This will give you the derivative of the given integral with respect to ( x ).

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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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