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

Question

Samantha Adkison · Accepted Answer

#20root(20)5# #int_0^5dx/x^(19/20)# Take the limit as #xrarr0#, since #1/x^(19/20)# is undefined for #x=0#. #=lim_(brarr0)int_b^5x^(-19/20)dx# #=lim_(brarr0)[20x^(1/20)]_b^5# #=lim_(brarr0)(20(5)^(1/20)-20(b)^(1/20))# Note that #20(0)^(1/20)=0#: #=20(5)^(1/20)# #=20root(20)5#

Adam Adkins · Answer

undefined 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.