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How do you use the second fundamental theorem of Calculus to find the derivative of given #int (t^2+1)^20 dt# from #[1,x]#?

Answer 1

Isaiah Allen

#= (x^2+1)^20 #

By the Second Fundamental Theorem of Calculus:

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

so:#d/dx int_0^x (t^2+1)^20 dt#

#= (x^2+1)^20 #

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Answer 2

Adam Adkins

To find the derivative of the integral (\int_{1}^{x} (t^2+1)^{20} dt) with respect to (x), you can use the second fundamental theorem of calculus, which states that if (f(t)) is a continuous function on ([a, b]) and (F(x) = \int_{a}^{x} f(t) dt), then (F'(x) = f(x)).

Given the integral (\int_{1}^{x} (t^2+1)^{20} dt), let (F(x) = \int_{1}^{x} (t^2+1)^{20} dt). According to the second fundamental theorem of calculus, (F'(x) = (x^2+1)^{20}). Therefore, the derivative of the given integral with respect to (x) is ((x^2+1)^{20}).

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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.