What is #f(x) = int sqrt(x+8) dx# if #f(2)=-3 #?

Answer 1

I got: #f(x)=2/3sqrt((x+8)^3)-(9+20sqrt(10))/3#

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Answer 2

To find ( f(x) = \int \sqrt{x+8} , dx ) given that ( f(2) = -3 ), you need to find the antiderivative of ( \sqrt{x+8} ), then evaluate it at ( x = 2 ) and use the given value to solve for the constant of integration.

First, find the antiderivative of ( \sqrt{x+8} ):

[ \int \sqrt{x+8} , dx = \frac{2}{3} (x + 8)^{3/2} + C ]

Given that ( f(2) = -3 ), substitute ( x = 2 ) into the antiderivative:

[ -3 = \frac{2}{3} (2 + 8)^{3/2} + C ]

[ -3 = \frac{2}{3} (10)^{3/2} + C ]

[ -3 = \frac{2}{3} \times 10 \times \sqrt{10} + C ]

[ -3 = \frac{20}{3} \sqrt{10} + C ]

[ C = -3 - \frac{20}{3} \sqrt{10} ]

Now, substitute the value of ( C ) back into the antiderivative:

[ f(x) = \frac{2}{3} (x + 8)^{3/2} - 3 - \frac{20}{3} \sqrt{10} ]

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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.

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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