How do you find the definite integral of #int x(x^(2/3))dx # from #[0,1]#?
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To find the definite integral of ( \int_0^1 x(x^{2/3}) , dx ), you can follow these steps:

Use the power rule for integration to integrate ( x(x^{2/3}) ). [ \int x(x^{2/3}) , dx = \int x^{5/3} , dx ]

Apply the power rule for integration, which states that ( \int x^n , dx = \frac{x^{n+1}}{n+1} + C ). [ \int x^{5/3} , dx = \frac{3}{8}x^{8/3} + C ]

Evaluate the definite integral by substituting the upper and lower limits of integration (1 and 0, respectively) into the antiderivative and subtracting the results. [ \int_0^1 x(x^{2/3}) , dx = \left[\frac{3}{8}x^{8/3}\right]_0^1 ] [ = \left(\frac{3}{8}(1)^{8/3}\right)  \left(\frac{3}{8}(0)^{8/3}\right) ] [ = \frac{3}{8}  0 ] [ = \frac{3}{8} ]
Therefore, the definite integral of ( \int_0^1 x(x^{2/3}) , dx ) is ( \frac{3}{8} ).
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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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