# What is #f(x) = int -9x+5sqrt(x^2+1) dx# if #f(2) = 7 #?

Separate the integrals.

This is a known integral that can be found here.

We now put all of this together and add the constant of integration.

Hopefully this helps!

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To find ( f(x) = \int -9x + 5\sqrt{x^2 + 1} , dx ) given ( f(2) = 7 ):

First, integrate ( -9x + 5\sqrt{x^2 + 1} ) with respect to ( x ):

[ \int -9x + 5\sqrt{x^2 + 1} , dx = -\frac{9}{2}x^2 + \frac{5}{3}(x^2 + 1)^{3/2} + C ]

Next, plug in ( x = 2 ) and ( f(2) = 7 ) into the equation:

[ -\frac{9}{2}(2)^2 + \frac{5}{3}((2)^2 + 1)^{3/2} + C = 7 ]

Solve for ( C ):

[ C = 7 - \left( -\frac{9}{2}(2)^2 + \frac{5}{3}((2)^2 + 1)^{3/2} \right) ]

[ C = 7 - \left( -18 + \frac{5}{3}(5)^{3/2} \right) ]

[ C = 7 - (-18 + \frac{5}{3}(125)^{1/2}) ]

[ C = 7 - (-18 + \frac{5}{3}(125)^{1/2}) ]

[ C = 7 - (-18 + \frac{5}{3}(125)^{1/2}) ]

[ C ≈ 7 + 23.463 ]

[ C ≈ 30.463 ]

Therefore, the function ( f(x) = \int -9x + 5\sqrt{x^2 + 1} , dx ) with ( f(2) = 7 ) is approximately:

[ f(x) = -\frac{9}{2}x^2 + \frac{5}{3}(x^2 + 1)^{3/2} + 30.463 ]

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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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