# What is #f(x) = int xsqrt(3x) dx# if #f(3) = 0 #?

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To find the function ( f(x) = \int x\sqrt{3x} , dx ) when ( f(3) = 0 ), we can first find the antiderivative of ( x\sqrt{3x} ) and then evaluate it at ( x = 3 ).

First, let's find the antiderivative:

[ \int x\sqrt{3x} , dx ]

Let ( u = 3x ), then ( du = 3dx ) or ( dx = \frac{du}{3} ).

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

Using the power rule for integration:

[ \frac{1}{3} \cdot \frac{2}{5}u^{\frac{5}{2}} + C = \frac{2}{15}u^{\frac{5}{2}} + C ]

Substitute back ( u = 3x ):

[ \frac{2}{15}(3x)^{\frac{5}{2}} + C = \frac{2}{15} \cdot 3^{\frac{5}{2}}x^{\frac{5}{2}} + C ]

Now, we know that ( f(3) = 0 ), so we can plug in ( x = 3 ) and solve for ( C ):

[ \frac{2}{15} \cdot 3^{\frac{5}{2}} \cdot 3^{\frac{5}{2}} + C = 0 ] [ \frac{2}{15} \cdot 3^5 + C = 0 ] [ \frac{2}{15} \cdot 243 + C = 0 ] [ \frac{486}{15} + C = 0 ] [ C = -\frac{486}{15} ]

So, the function is:

[ f(x) = \frac{2}{15} \cdot 3^{\frac{5}{2}}x^{\frac{5}{2}} - \frac{486}{15} ]

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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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