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

Answer 1

Charles Anctil

#= (cos x) / x#

The 2nd FTC states that #d/dx int_a^x f(t) dt = f(x)#

So here #d/dx int_3^x cos(t) / t dt = (cos x) / x#

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Answer 2

Ezekiel Alexander

To use the Second Fundamental Theorem of Calculus to find the derivative of the integral of ( \int_{3}^{x} \frac{\cos(t)}{t} , dt ), you first evaluate the integral function ( F(x) = \int_{a}^{x} f(t) , dt ), where ( f(t) = \frac{\cos(t)}{t} ) and ( a = 3 ). Then, you differentiate ( F(x) ) with respect to ( x ) to find ( F'(x) ), which represents the derivative of the given integral function.

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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.