# How do you evaluate the integral #int dx/(x^(19/20))# from 0 to 5 if it converges?

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To evaluate the integral ( \int \frac{dx}{x^{19/20}} ) from 0 to 5, we first need to check if the integral converges. Since the integrand ( \frac{1}{x^{19/20}} ) is continuous and positive on the interval [0, 5], and the exponent ( \frac{19}{20} ) is greater than 1, the integral converges.

To find the value of the integral, we can compute it directly. The antiderivative of ( \frac{1}{x^{19/20}} ) is ( x^{-1/20} ). Thus, the definite integral is:

[ \int_0^5 \frac{dx}{x^{19/20}} = \left[ -20x^{-1/20} \right]_0^5 = -20(5^{-1/20}) - (-20(0^{-1/20})) ]

Since ( 0^{-1/20} ) is undefined, we need to evaluate the limit as ( x ) approaches 0:

[ \lim_{x \to 0^+} -20x^{-1/20} = -20 \cdot \lim_{x \to 0^+} \frac{1}{x^{1/20}} = -20 \cdot \infty = -\infty ]

Therefore, the integral from 0 to 5 diverges.

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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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