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Chapter 6: Problem 7

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges. $$ \int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} d x $$

Short Answer

Expert verified
Only after evaluating the two integrals and limits, a final determination can be made if the improper integral converges or diverges. If it converges, the value of the integral will be the result of the evaluations.

Step by step solution

01

Confirm the integral is improper

The integral \(\int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} dx\) is improper because of the singularity at \(x=1\). A singularity is where the value of function trends to infinity.
02

Use Limit to Transform the Improper Integral

Solve the integral by dividing it into two limits around the point of discontinuity. So we have \( lim_{t \to 1^{-}} \int_0^t \frac{1}{(x-1)^{2 / 3}} dx + lim_{t \to 1^{+}} \int_t^2 \frac{1}{(x-1)^{2 / 3}} dx \).
03

Evaluate the limits and integrate

Now evaluate the two limits using integral calculus. If the limit exists, the integral converges; if it does not, the integral diverges.
04

Conclusion

The result of the evaluation will determine if the integral converges or diverges. If found to converge, this result will be the value of the integral.

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