Integral of ln(x^2)/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(x2).
Then let du=x2dx and substitute 2du:
∫4udu
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The integral of a constant times a function is the constant times the integral of the function:
∫2udu=2∫udu
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The integral of un is n+1un+1 when n=−1:
∫udu=2u2
So, the result is: 4u2
Now substitute u back in:
4log(x2)2
Method #2
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Let u=x2.
Then let du=2xdx and substitute 2du:
∫4ulog(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫2ulog(u)du=2∫ulog(u)du
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Let u=u1.
Then let du=−u2du and substitute −du:
∫ulog(u1)du
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The integral of a constant times a function is the constant times the integral of the function:
∫(−ulog(u1))du=−∫ulog(u1)du
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Let u=log(u1).
Then let du=−udu and substitute −du:
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The integral of a constant times a function is the constant times the integral of the function:
∫(−u)du=−∫udu
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The integral of un is n+1un+1 when n=−1:
∫udu=2u2
So, the result is: −2u2
Now substitute u back in:
−2log(u1)2
So, the result is: 2log(u1)2
Now substitute u back in:
2log(u)2
So, the result is: 4log(u)2
Now substitute u back in:
4log(x2)2
Method #3
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Let u=x1.
Then let du=−x2dx and substitute −du:
∫ulog(u21)du
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The integral of a constant times a function is the constant times the integral of the function:
∫(−ulog(u21))du=−∫ulog(u21)du
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Let u=log(u21).
Then let du=−u2du and substitute −2du:
∫4udu
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The integral of a constant times a function is the constant times the integral of the function:
∫(−2u)du=−2∫udu
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The integral of un is n+1un+1 when n=−1:
∫udu=2u2
So, the result is: −4u2
Now substitute u back in:
−4log(u21)2
So, the result is: 4log(u21)2
Now substitute u back in:
4log(x2)2
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Add the constant of integration:
4log(x2)2+constant
The answer is:
4log(x2)2+constant
The answer (Indefinite)
[src]
/
|
| / 2\ 2/ 2\
| log\x / log \x /
| ------- dx = C + --------
| x 4
|
/
(logx)2
Use the examples entering the upper and lower limits of integration.