# Find the derivative of #int_0^(x^3e^x) \(t^3+3)^17 \ dt#?

Because we need the chain rule.

and

so

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# d/dx int_0^(x^3e^x) \ (t^3+3)^17 \ dt = (x^3e^x + 3x^2e^x)((x^3e^x)^3+3)^17#

If asked to find the derivative of an integral then you should not evaluate the integral but instead use the fundamental theorem of Calculus.

The FTOC tells us that:

(ie the derivative of an integral gives us the original function back).

We are asked to find:

(notice the upper bounds of the first integral are not in the correct format for the FTOC to be applied, directly). We can manipulate the definite integral using a substitution the product rule, and the chain rule. Let:

The substituting into the integral [A], and applying the chain rule, we get:

And now the derivative of the integral is in the correct form for the FTOC to be applied, giving:

And restoring the initial substitution we get:

Which is option (2) in the initial question.

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The derivative of ( \int_{0}^{x^3e^x} (t^3+3)^{17} , dt ) withTo find the derivative of the given integral with respect toThe derivative of ( \int_{0}^{x^3e^x} (t^3+3)^{17} , dt ) with respect toTo find the derivative of the given integral with respect to ( x \The derivative of ( \int_{0}^{x^3e^x} (t^3+3)^{17} , dt ) with respect to (To find the derivative of the given integral with respect to ( x ), we apply the Fundamental TheoremThe derivative of ( \int_{0}^{x^3e^x} (t^3+3)^{17} , dt ) with respect to ( x ) is ( (x^3e^x)^3(e^x + 3)^{17} ).To find the derivative of the given integral with respect to ( x ), we apply the Fundamental Theorem of Calculus and the Chain Rule:

[ \frac{d}{dx} \left( \int_0^{x^3e^x} (t^3 + 3)^{17} dt \right) = \frac{d}{dx} \left( F(x^3e^x) \right) ]

where ( F(x) ) is the antiderivative of ( (t^3 + 3)^{17} ). Applying the Chain Rule:

[ \frac{d}{dx} \left( F(x^3e^x) \right) = \frac{dF}{du} \cdot \frac{du}{dx} ]

where ( u = x^3e^x ). Now, find ( \frac{dF}{du} ) and ( \frac{du}{dx} ):

[ \frac{dF}{du} = (u^3 + 3)^{17} \cdot 3u^2 \cdot e^x ] [ \frac{du}{dx} = (3x^2e^x + x^3e^x) ]

Thus, the derivative is:

[ \frac{d}{dx} \left( \int_0^{x^3e^x} (t^3 + 3)^{17} dt \right) = (x^3e^x)^3 + 3)^{17} \cdot 3(x^3e^x)^2 \cdot e^x \cdot (3x^2e^x + x^3e^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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