What is #f(x) = int xsqrt(5-2) dx# if #f(2) = 3 #?

Answer 1

The function #f(x)# is #(x^2sqrt3+6-4sqrt3)/2#.

First, compute the integral:

#f(x)=intxsqrt(5-2)# #dx#
#color(white)(f(x))=intxsqrt3# #dx#
#color(white)(f(x))=sqrt3intx# #dx#

Power rule:

#color(white)(f(x))=sqrt3*x^2/2+C#
Now, set #f(2)# (which is #3#) and the integral evaluated at #2# equal to each other, then solve for #C#:
#3=sqrt3*2^2/2+C#
#3=2sqrt3+C#
#3-2sqrt3=C#
That's the #C# value, so that means that the function is:
#f(x)=sqrt3*x^2/2+3-2sqrt3#

If you would like, you can rewrite it as:

#f(x)=(x^2sqrt3+6-4sqrt3)/2#

Hope this helped!

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Answer 2

To find ( f(x) = \int x \sqrt{5 - 2x} , dx ) when ( f(2) = 3 ), we need to find the definite integral of the given function and then use the given value to determine the constant of integration.

Since ( f(2) = 3 ), this means:

[ \int_{2}^{2} x \sqrt{5 - 2x} , dx = 3 ]

This integral evaluates to zero since the lower and upper limits of integration are the same.

Now, we have:

[ \int_{2}^{x} x \sqrt{5 - 2x} , dx = f(x) - f(2) ]

We need to evaluate this integral to find ( f(x) ). Then we can use the given information to find the value of ( f(x) ).

After evaluating the integral, we find:

[ f(x) = \frac{5}{12} (5 - 2x)^{3/2} + C ]

Now, we use the given value ( f(2) = 3 ) to find the constant of integration ( C ).

[ 3 = \frac{5}{12} (5 - 2(2))^{3/2} + C ]

[ 3 = \frac{5}{12} (5 - 4)^{3/2} + C ]

[ 3 = \frac{5}{12} (1)^{3/2} + C ]

[ 3 = \frac{5}{12} + C ]

[ C = 3 - \frac{5}{12} ]

[ C = \frac{31}{12} ]

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

[ f(x) = \frac{5}{12} (5 - 2x)^{3/2} + \frac{31}{12} ]

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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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