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By using intergration by parts, show that the intergral `int_2^4 X( ln X )dx` = `a (ln...

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By using intergration by parts, show that the intergral

`int_2^4 X( ln X )dx` = `a (ln b)` `+ c`

Where a,b and c are integers to be determined.

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Let;

`U = x^2/2`

`dU = xdx`

 

`V = lnx`

`dV = 1/xdx`

 

Using integral by parts;

`intVdU = UV-intUdV`

 

`int_2^4lnx*xdx = [x^2/2*lnx]_2^4-int_2^4(x^2/2)*1/xdx`

`int_2^4xlnxdx = [x^2/2*lnx]_2^4-int_2^4(x/2)dx`

`int_2^4xlnxdx = [x^2/2*lnx]_2^4-[x^2/4]_2^4`

`int_2^4xlnxdx = (4^2/2xxln4-2^2/2xxln2)-(4^2/4-2^2/4)`

`int_2^4xlnxdx = (16ln2-2ln2)-(4-1)`

`int_2^4xlnxdx = 14ln2-3`

 

This looks the form of `alnb+c` where;

a = 14

b = 2

c = -3

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