Prove int_1^3x/(2x-1)^3dx = 8/25 , Using substitution u = 2x - 1 ?

1 Answer
Mar 8, 2018

See the explanation below

Explanation:

As suggested, let

u=2x-1, =>, du=2dx

2x=u+1

x=(u+1)/2

Start by calculating the indefinite integral

Therefore,

int(xdx)/(2x-1)^3=1/4int((u+1)du)/u^3

=1/4int(1/u^2+1/u^3)du

=1/4(-1/u-1/(2u^2))

=1/4(-1/(2x-1)-1/(2(2x-1)^2))+C

Now, calculate the definite integral

int_1^3(xdx)/(2x-1)^3=[-1/4*1/(2x-1)-1/8*1/(2x-1)^2)]_1^3

=(-1/20-1/200)-(-1/4-1/8)

=3/8-11/200

=64/200

=8/25