How do you find the integration of \log x

gorovogpg

gorovogpg

Answered question

2021-12-12

How do you find the integration of logx

Answer & Explanation

Jack Maxson

Jack Maxson

Beginner2021-12-13Added 25 answers

Explanation:
log(x)dx=ln(x)ln(10)dx
=1ln(10)ln(x)dxln(x)dx
Using the integration by parts :
f(x)g(x)dx=[f(x)g(x)]f(x)g(x)dx
There: f(x)=ln(x),f(x)=1x,g(x)=x,g(x)=1
So: log(x)dx=1ln(10)(xln(x)dx)
So: log(x)dx=1ln(10)(xln(x)x)+C=xln(10)(ln(x)1)+C
In general, logn(x)dx=xln(n)(ln(x)x)+C=xln(10)(ln(x)1)+C
In general, logn(x)dx=xln(n)(ln(x)1)+C
Jillian Edgerton

Jillian Edgerton

Beginner2021-12-14Added 34 answers

Explanation:
Remember that:
loga(b)=logc(b)logc(a)
log10(x)=ln(x)ln(10)
We now have:
ln(x)1ln(10)dx
1ln(10)ln(x)dx
Integration by parts:
udv=uvvdu
We let:
u=ln(x)
dv=1
du=ddx(ln(x))
du=1x
v=1dx
v=x
1ln(10)[xln(x)x1xdx]
1ln(10)[xln(x)1dx]
1ln(10)[xln(x)x]
xln(x)xln(10)
xln(x)xln(10)
x(ln(x)1)ln(10) Do you C why this is incomplete?
log(x)dx=x(ln(x)1)ln(10)+C

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