Using substitution the integral from 2 to 4 dx/xlnx

Question

While using substitution method how would I go about integrating dx/xlnx?

Answer 1

This can easily be done with integration by parts, if you've learned that.

 

∫udv = uv-∫vdu

 

For ∫lnx, your u would need to be the lnx.  Since there is nothing else there, v could be x, so that dv=1

 

u=lnx, du=1/x

v=x, dv=1

 

• ∫lnx = xlnx-∫1dx  (This is now easily integrated)
• xlnx-x+C
• (Remember this is all now from 2 to 4)
• 4ln4-4-2ln2+2
• 4ln4-2ln2-2
• approx 2.1589

 

 

Answer 2

Find the indefinite integral first, then add the boundary conditions.

 

For this problem, we want to use integration by parts method.

 

integral(uv) = uv - integral(vdu) , where v and u are functions.

 

Let dv be a function in which it is easily integrated.

Le u be a function in which you can derive.

 

u = lnx                  dv = xdx

du =x^-1dx                v = (1/2)x²