# How do you use the Fundamental Theorem of Calculus to find the derivative of #int 5(sin(t))^5 dt# from 5 to e^x?

here a = 5, u = e^x

so FTC + chain rules tells us that

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To find the derivative of the integral ( \int_{5}^{e^x} \sin(t)^5 , dt ), you can use the Fundamental Theorem of Calculus. According to the theorem, if ( F(x) ) is any antiderivative of ( f(x) ), then ( \frac{d}{dx} \int_{a}^{x} f(t) , dt = f(x) ). In this case, ( f(t) = \sin(t)^5 ).

First, find an antiderivative of ( f(t) ), which is ( F(t) ). Then evaluate ( F(e^x) - F(5) ). Finally, take the derivative of this expression with respect to ( x ).

Let's find ( F(t) ): [ F(t) = \int \sin(t)^5 , dt ]

Now, we find ( F(e^x) - F(5) ): [ F(e^x) - F(5) = \left[ \int \sin(t)^5 , dt \right]_{t=5}^{t=e^x} ]

Finally, take the derivative of this expression with respect to ( x ). 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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