`int_3^4 1/(x-3)^(3/2) dx` Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converges

Expert Answers

An illustration of the letter 'A' in a speech bubbles

The integral is improper because the function under the integral `f(x)=1/(x-3)^(3/2)` is not defined at 3 (for `x=3` denominator is equal to zero). 

`int_3^4 1/(x-3)^(3/2)dx=`

Substitute `u=x-3` `=>` `du=dx,` `u_l=3-3=0,` `u_u=4-3=1.`

`u_l` and `u_u` denote new lower and upper bound respectively.

`int_0^1 1/u^(3/2)du=int_0^1 u^(-3/2)du=-2u^(-1/2)|_0^1=-2cdot1^(-1/2)+lim_(u to 0)2u^(-1/2)=`


As we can see the integral diverges.

The image below shows the graph of the function and area under it corresponding to the integral. Both axis are asymptotes of the function.                                                                                                                   

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Image (1 of 1)
Approved by eNotes Editorial