# 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

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)=

-2+infty=infty

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.

Images:
This image has been Flagged as inappropriate Click to unflag
Image (1 of 1)
Approved by eNotes Editorial Team