use integration by parts to integrate integration sign, ln x / square root x, whole equation by dx

Asked on by jmg5639

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beckden | High School Teacher | (Level 1) Educator

Posted on

int ln(x)/sqrt(x) dx

use LIPET to find to use `u=ln(x)` , `dv = 1/sqrt(x) dx`

This gives `du = 1/x dx` and `v = 2sqrt(x)`

Now apply integration by parts

`int udv = uv - int v du`   to get

`int ln(x)/sqrt(x) dx = ln(x) (2sqrt(x)) - int 2sqrt(x) 1/x dx`

`int ln(x)/sqrt(x) dx = 2sqrt(x)ln(x) - 2 int sqrt(x)/x dx`

Since `sqrt(x)/x = x^(1/2)/x^1 = x^(-1/2)`

`int ln(x)/sqrt(x) dx = 2sqrt(x)ln(x) - 2 int x^(-1/2) dx`

And finally we get

`int ln(x)/sqrt(x) dx = 2sqrt(x)ln(x) - 4sqrt(x) + C` which is the same as the answer above but without using a table of integrals.

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jeew-m | College Teacher | (Level 1) Educator Emeritus

Posted on

Let x=t^2

then by differenciation;

   1= 2t*dt/dx

dx = 2tdt


`int(lnx/sqrtx)dx = int(lnt^2/sqrtt^2)*2tdt`

                = `int(2lnt)/t*2tdt`

                =  `4intlnt dt`


                = `4[sqrtx*lnsqrtx-sqrtx]+C`

                =` 4sqrtx(lnsqrtx-1)+C`


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