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How do you use the Fundamental Theorem of Calculus to find the derivative of #int e^(4t²-t) dt# from 3 to x?

Answer 1

Charles Arocha

By FTC

#d/(dx) int_a^x \ f(t) \ dt = f(x)#

#d/dx int_3^x e^(4t²-t) dt#

#=e^(4x²-x) #

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Answer 2

Cameron Alexander

To find the derivative of the integral ( \int_{3}^{x} e^{4t^2 - t} , dt ), you can use the Fundamental Theorem of Calculus. The Fundamental Theorem of Calculus states that if ( F(x) ) is the antiderivative of ( f(x) ), then ( \int_{a}^{x} f(t) , dt = F(x) - F(a) ).

In this case, ( f(t) = e^{4t^2 - t} ). To find the antiderivative ( F(x) ), integrate ( f(t) ) with respect to ( t ):

[ F(x) = \int e^{4t^2 - t} , dt ]

After finding ( F(x) ), you can differentiate ( F(x) ) with respect to ( x ) to find ( F'(x) ), which will be the derivative of the given integral:

[ F'(x) = \frac{d}{dx} \left( \int_{3}^{x} e^{4t^2 - t} , dt \right) ]

This process involves calculating the antiderivative of ( f(t) ), then finding its derivative with respect to ( x ).

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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.