How do you evaluate the definite integral #int (xsqrtx)dx# from [4,9]?
The answer is
To evaluate this integral, we use
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To evaluate the definite integral ( \int_{4}^{9} x \sqrt{x} , dx ), we can use the integration by substitution method. Let ( u = \sqrt{x} ). Then, ( x = u^2 ) and ( dx = 2u , du ). Substituting these into the integral, we get:
[ \int_{4}^{9} x \sqrt{x} , dx = \int_{2}^{3} u^2 \cdot u \cdot 2u , du ]
[ = 2 \int_{2}^{3} u^4 , du ]
[ = 2 \left[ \frac{u^5}{5} \right]_{2}^{3} ]
[ = 2 \left( \frac{3^5}{5}  \frac{2^5}{5} \right) ]
[ = 2 \left( \frac{243  32}{5} \right) ]
[ = 2 \left( \frac{211}{5} \right) ]
[ = \frac{422}{5} ]
So, the value of the definite integral ( \int_{4}^{9} x \sqrt{x} , dx ) is ( \frac{422}{5} ).
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To evaluate the definite integral (\int_{4}^{9} x \sqrt{x} , dx), you can use the power rule for integration and the properties of definite integrals. First, rewrite the integrand as (x^{\frac{3}{2}}), then integrate using the power rule. Finally, evaluate the antiderivative at the upper and lower limits of integration and find the difference. Here's the calculation:

Integrate (x^{\frac{3}{2}}) with respect to (x): [ \int x^{\frac{3}{2}} , dx = \frac{2}{5} x^{\frac{5}{2}} + C ]

Evaluate the antiderivative at the upper and lower limits: [ \left(\frac{2}{5} \cdot 9^{\frac{5}{2}}\right)  \left(\frac{2}{5} \cdot 4^{\frac{5}{2}}\right) ]

Simplify and compute the difference to find the value of the definite integral.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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