# How do you find the integral of 0 to the infinity of #x^(8/3) dx#?

The integral diverges.

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To find the integral of ( \int_0^\infty x^{\frac{8}{3}} , dx ), you can use the technique of improper integration. First, evaluate the indefinite integral ( \int x^{\frac{8}{3}} , dx ), which equals ( \frac{3}{11}x^{\frac{11}{3}} + C ). Then, apply the limits of integration from 0 to ( \infty ).

To handle the infinity limit, you evaluate the integral from 0 to a finite value ( b ), and then take the limit as ( b ) approaches infinity.

So, the integral becomes ( \lim_{b \to \infty} \int_0^b x^{\frac{8}{3}} , dx ).

Substitute the limits and integrate:
[ \lim_{b \to \infty} \left[ \frac{3}{11}x^{\frac{11}{3}} \right]*0^b ]
[ = \lim*{b \to \infty} \left( \frac{3}{11}b^{\frac{11}{3}} - \frac{3}{11}(0)^{\frac{11}{3}} \right) ]
[ = \lim_{b \to \infty} \left( \frac{3}{11}b^{\frac{11}{3}} - 0 \right) ]
[ = \lim_{b \to \infty} \frac{3}{11}b^{\frac{11}{3}} ]

As ( b ) approaches infinity, ( b^{\frac{11}{3}} ) also approaches infinity. Therefore, the limit is infinity.

So, the integral ( \int_0^\infty x^{\frac{8}{3}} , dx ) diverges to infinity.

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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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