How do you test the improper integral #int (2x-1)^(-2/3)dx# from #[0,1]# and evaluate if possible?

Answer 1

The integral converges to #=3#

The critical point is when #x=1/2#

Therefore,

The integral is

#int_0^1(2x-1)^(-2/3)dx #
#= lim_(t->(1/2))int_0^t(dx)/(2x-1)^(2/3)+lim_(t->(1/2))int_t^1(dx)/(2x-1)^(2/3)#
#= lim_(t->(1/2))[3/2(2x-1)^(1/3)]_0^t+ lim_(t->(1/2))[3/2(2x-1)^(1/3)]_t^1#
#= lim_(t->(1/2))(3/2*(2t-1)^(1/3)-3/2*(-1)^(1/3)) + lim_(t->(1/2))(3/2*(1)^(1/3)-3/2*(2t-1)^(1/3)) #
#=3/2+3/2#
#=3#
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Answer 2

To test the improper integral (\int_{0}^{1} (2x-1)^{-\frac{2}{3}} , dx) from (0) to (1), we first check if it exists by examining the behavior of the integrand as it approaches the limits of integration. Specifically, we analyze the behavior as (x) approaches the endpoints (0) and (1).

At (x = 0), the integrand becomes ((2(0)-1)^{-\frac{2}{3}} = (-1)^{-\frac{2}{3}}). Since this results in an indeterminate form (division by zero), we must evaluate the limit:

[\lim_{x \to 0^+} (2x-1)^{-\frac{2}{3}}]

Similarly, at (x = 1), the integrand becomes ((2(1)-1)^{-\frac{2}{3}} = (1)^{-\frac{2}{3}}). Again, we evaluate the limit:

[\lim_{x \to 1^-} (2x-1)^{-\frac{2}{3}}]

If both limits exist and are finite, then the improper integral converges. Otherwise, it diverges.

After confirming the existence of the improper integral, we proceed to evaluate it by finding its antiderivative, then applying the Fundamental Theorem of Calculus to compute the definite integral over the interval ([0,1]). However, the antiderivative of ((2x-1)^{-\frac{2}{3}}) does not have a simple elementary form, so numerical or approximate methods may be needed to evaluate the integral.

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Answer from HIX Tutor

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.

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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