Determine whether the integral is divergent or convergent. integrate from 2 to 8 of (1)/((x-6)^3)dx If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF. If it...
Determine whether the integral is divergent or convergent.
integrate from 2 to 8 of (1)/((x-6)^3)dx
If it is convergent, evaluate it. If it diverges to infinity, state your answer as INF. If it diverges to negative infinity, state your answer as MINF. If it diverges without being infinity or negative infinity, state your answer as DIV.
- print Print
- list Cite
Expert Answers
calendarEducator since 2012
write738 answers
starTop subjects are Math and Science
To determine if the integral is convergent or divergent, we need to evaluate the integral up to the vertical asymptote of the integrand and then take the limit to the asymptote.
`int_2^8 1/(x-6)^3dx` let `u=x-6` then `du=dx` and the limits become -4 and 2
`=int_{-4}^2 u^{-3}du`
`=lim_{epsilon->0}(int_{-4}^epsilon u^{-3}du)+lim_{epsilon->0}(int_epsilon^2 u^{-3}du)`
`=lim_{epsilon->0}(-1/2u^{-2}|_{-4}^epsilon)+lim_{epsilon->0}(-1/2 u^{-2}|_epsilon^2)`
`=-1/2lim_{epsilon->0}(1/epsilon^2-1/16)-1/2lim_{epsilon->0}(1/4-1/epsilon^2)`
Since each part is divergent to negative infinity, the integral is divergent to negative infinity (MINF).
Related Questions
- Determine whether the integral is divergent or convergent. integrate from 2 to 8 of...
- 2 Educator Answers
- Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If...
- 1 Educator Answer
- integrate from 5 to infinity of (ln(2x))/(x)dx Is the integral is divergent or convergent? If it...
- 1 Educator Answer
- Determine which values of p the following integral converge. Give your answer in each case by...
- 1 Educator Answer
- Determine whether the sequence converges or diverges. If it converges, find the limit. (If an...
- 1 Educator Answer
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.