Integral of ln(x)/(x^2) dx
The solution
Detail solution
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There are multiple ways to do this integral.
Method #1
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Let u=log(x).
Then let du=xdx and substitute du:
∫ue−udu
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Let u=−u.
Then let du=−du and substitute du:
∫ueudu
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Use integration by parts:
∫udv=uv−∫vdu
Let u(u)=u and let dv(u)=eu.
Then du(u)=1.
To find v(u):
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The integral of the exponential function is itself.
∫eudu=eu
Now evaluate the sub-integral.
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The integral of the exponential function is itself.
∫eudu=eu
Now substitute u back in:
−ue−u−e−u
Now substitute u back in:
−xlog(x)−x1
Method #2
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=log(x) and let dv(x)=x21.
Then du(x)=x1.
To find v(x):
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The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
Now evaluate the sub-integral.
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The integral of a constant times a function is the constant times the integral of the function:
∫(−x21)dx=−∫x21dx
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The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
So, the result is: x1
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Now simplify:
−xlog(x)+1
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Add the constant of integration:
−xlog(x)+1+constant
The answer is:
−xlog(x)+1+constant
The answer (Indefinite)
[src]
/
|
| log(x) 1 log(x)
| ------ dx = C - - - ------
| 2 x x
| x
|
/
∫x2log(x)dx=C−xlog(x)−x1
Use the examples entering the upper and lower limits of integration.