# What is #f(x) = int x+3xsqrt(x^2+1) dx# if #f(2) = 7 #?

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The integral of (x + 3x\sqrt{x^2 + 1}) with respect to (x) is ( \frac{x^2}{2} + \frac{3}{5}(x^2 + 1)^{\frac{3}{2}} + C).

Given that (f(2) = 7), we can solve for (C):

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

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

[7 = 2 + \frac{3}{5}(5\sqrt{5}) + C]

[7 = 2 + \frac{3}{5}(5\sqrt{5}) + C]

[7 = 2 + 3\sqrt{5} + C]

[C = 7 - 2 - 3\sqrt{5}]

[C = 5 - 3\sqrt{5}]

Therefore, the function (f(x)) is:

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

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The function ( f(x) = \int (x + 3x\sqrt{x^2 + 1}) , dx ) represents the integral of the expression ( (x + 3x\sqrt{x^2 + 1}) ) with respect to ( x ). Given that ( f(2) = 7 ), we need to find the value of the integral when ( x = 2 ).

First, we need to find the antiderivative of the expression inside the integral:

[ \int (x + 3x\sqrt{x^2 + 1}) , dx ]

[ = \int x , dx + \int 3x\sqrt{x^2 + 1} , dx ]

[ = \frac{x^2}{2} + \int 3x\sqrt{x^2 + 1} , dx ]

To find ( \int 3x\sqrt{x^2 + 1} , dx ), we can use substitution. Let ( u = x^2 + 1 ), then ( du = 2x , dx ), or ( \frac{du}{2} = x , dx ).

[ \int 3x\sqrt{x^2 + 1} , dx = \int 3\sqrt{u} , \frac{du}{2} ]

[ = \frac{3}{2} \int \sqrt{u} , du ]

[ = \frac{3}{2} \cdot \frac{2}{3} u^{3/2} + C ]

[ = u^{3/2} + C ]

[ = (x^2 + 1)^{3/2} + C ]

Therefore, the antiderivative of ( f(x) ) is ( F(x) = \frac{x^2}{2} + (x^2 + 1)^{3/2} + C ), where ( C ) is the constant of integration.

Now, we can find the value of ( F(2) ):

[ F(2) = \frac{2^2}{2} + (2^2 + 1)^{3/2} + C ]

[ = 2 + (5)^{3/2} + C ]

Given that ( f(2) = 7 ), we can solve for ( C ):

[ 7 = 2 + (5)^{3/2} + C ]

[ C = 7 - 2 - (5)^{3/2} ]

[ C = 7 - 2 - 5\sqrt{5} ]

[ C = 5 - 5\sqrt{5} ]

Therefore, the value of the integral ( f(x) ) when ( x = 2 ) is ( 2 + (5)^{3/2} + 5 - 5\sqrt{5} ).

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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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