Integral of ln(ln(x))/x 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(log(x)).
Then let du=xlog(x)dx 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:
log(x)log(log(x))−log(x)
Method #2
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Let u=log(x).
Then let du=xdx and substitute du:
∫log(u)du
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Use integration by parts:
∫udv=uv−∫vdu
Let u(u)=log(u) and let dv(u)=1.
Then du(u)=u1.
To find v(u):
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The integral of a constant is the constant times the variable of integration:
∫1du=u
Now evaluate the sub-integral.
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The integral of a constant is the constant times the variable of integration:
∫1du=u
Now substitute u back in:
log(x)log(log(x))−log(x)
Method #3
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Let u=x1.
Then let du=−x2dx and substitute −du:
∫(−ulog(log(u1)))du
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The integral of a constant times a function is the constant times the integral of the function:
∫ulog(log(u1))du=−∫ulog(log(u1))du
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Let u=log(log(u1)).
Then let du=−ulog(u1)du and substitute −du:
∫(−ueu)du
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The integral of a constant times a function is the constant times the integral of the function:
∫ueudu=−∫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):
-
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
So, the result is: −ueu+eu
Now substitute u back in:
−log(u1)log(log(u1))+log(u1)
So, the result is: log(u1)log(log(u1))−log(u1)
Now substitute u back in:
log(x)log(log(x))−log(x)
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Now simplify:
(log(log(x))−1)log(x)
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Add the constant of integration:
(log(log(x))−1)log(x)+constant
The answer is:
(log(log(x))−1)log(x)+constant
The answer (Indefinite)
[src]
/
|
| log(log(x))
| ----------- dx = C - log(x) + log(x)*log(log(x))
| x
|
/
∫xlog(log(x))dx=C+log(x)log(log(x))−log(x)
(122.846251720628 + 138.514221668049j)
(122.846251720628 + 138.514221668049j)
Use the examples entering the upper and lower limits of integration.