evaluate the improper interval with an interior discontinuity
dx/x^3 from -1 to 2
integral ** not interval
dx/x^3 is just x^-3 dx
so you just use the power rule:
∫dx/x^3 = -1/(2x^2)
Now we run into a snag, since -1/(2x^2) is undefined at x=0
So, we have to evaluate the limits from left and right.
Unfortunately, they also do not exist, so the integral does not converge.
To evaluate the improper integral with an interior discontinuity of the function dx/x^3 from -1 to 2, we need to split the interval at the point of the discontinuity (if any) and then evaluate each part separately.
In this case, the function dx/x^3 has an interior discontinuity at x = 0 since it is not defined for x = 0. Therefore, we split the interval [-1, 2] into two parts: [-1, 0) and (0, 2].
Let's first evaluate the improper integral over the interval [-1, 0).
∫(from -1 to 0) dx/x^3
Since x = 0 is the point of discontinuity, the integral is improper at x = 0. We can rewrite the integral as a limit:
∫(from -1 to 0) dx/x^3 = lim(a→0-) ∫(from -1 to a) dx/x^3
To evaluate this integral, we can use the power rule of integration:
∫ x^n dx = (x^(n+1))/(n+1) + C
Applying the power rule to our integral, we get:
lim(a→0-) [(x^(-2))/(-2)] (evaluated from -1 to a)
= lim(a→0-) [(-1/a^2 - (-1/(-1)^2))/(-2)]
= lim(a→0-) [(-1/a^2 + 1)/(-2)]
= lim(a→0-) [(1 - a^2)/(2a^2)]
Now, let's evaluate the improper integral over the interval (0, 2].
∫(from 0 to 2) dx/x^3
Since x = 0 is still the point of discontinuity, we rewrite the integral as a limit:
∫(from 0 to 2) dx/x^3 = lim(b→0+) ∫(from b to 2) dx/x^3
Using the power rule of integration, we get:
lim(b→0+) [(x^(-2))/(-2)] (evaluated from b to 2)
= lim(b→0+) [(-1/2)*(2^(-2) - b^(-2))]
= lim(b→0+) [(-1/2)*(1/4 - 1/b^2)]
= lim(b→0+) [(-1/2)*(b^2 - 4)/(4b^2)]
Next, we take the limit as both a and b approach 0 to find the value of the improper integral. Note that we cannot simply add the limits since the limits approach 0 from different directions (negative and positive).
lim(a→0-) [(1 - a^2)/(2a^2)] + lim(b→0+) [(-1/2)*(b^2 - 4)/(4b^2)]
Based on the limit, you can calculate the value of the improper integral.