# How do I evaluate #d/dx \int_5^(x^4) \sqrt{t^2 + t} dt#?

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Here comes the chaining part.

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To evaluate ( \frac{d}{dx} \int_5^{x^4} \sqrt{t^2 + t} , dt ), apply the Fundamental Theorem of Calculus and the Chain Rule. First, find the antiderivative of ( \sqrt{t^2 + t} ), then differentiate it with respect to ( x ). The result is:

[ \frac{d}{dx} \int_5^{x^4} \sqrt{t^2 + t} , dt = \frac{d}{dx} \left[ F(x^4) - F(5) \right] = 4x^3 \sqrt{(x^4)^2 + x^4} ]

where ( F(x) ) is an antiderivative of ( \sqrt{t^2 + t} ).

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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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