# How do you evaluate #d /dx \int_[x]^{x^4} sqrt{t^2+t} dt#?

# d/dx \ int_x^(x^4) \ sqrt(t^2+t) \ dt = 4x^3sqrt(x^8+x^4) - sqrt(x^2+x) #

You shouldn't evaluate an integral if you are asked to find its derivative using the calculus fundamental theorem.

According to the Calculus Fundamental Theorem:

(i.e., we can return to the original function using the derivative of an integral).

It is requested of us to locate:

We can manipulate the definite integral as follows (notice the upper and lower bounds are not in the correct format for the FTOC to be applied, directly):

The chain rule also allows us to write:

When we combine these unimportant findings, we obtain:

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To evaluate ( \frac{d}{dx} \int_{x}^{x^4} \sqrt{t^2 + t} , dt ), we apply the Fundamental Theorem of Calculus and the Chain Rule.

First, we find the antiderivative of ( \sqrt{t^2 + t} ) with respect to ( t ). Let ( F(t) ) be the antiderivative. Then,

[ F(t) = \int \sqrt{t^2 + t} , dt ]

Next, we evaluate ( F(x^4) - F(x) ).

[ \frac{d}{dx} \int_{x}^{x^4} \sqrt{t^2 + t} , dt = F'(x^4) \cdot (4x^3) - F'(x) \cdot 1 ]

This simplifies to:

[ 4x^3 \cdot \sqrt{(x^4)^2 + x^4} - \sqrt{x^2 + x} ]

[ = 4x^3 \cdot \sqrt{x^8 + x^4} - \sqrt{x^2 + x} ]

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