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Integral of log(2x^5)/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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Rewrite the integrand:
x2log(2x5)=x2log(x5)+log(2)
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Rewrite the integrand:
x2log(x5)+log(2)=x2log(x5)+x2log(2)
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Integrate term-by-term:
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Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=log(x5) and let dv(x)=x21.
Then du(x)=x5.
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:
∫(−x25)dx=−5∫x21dx
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The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
So, the result is: x5
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The integral of a constant times a function is the constant times the integral of the function:
∫x2log(2)dx=log(2)∫x21dx
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The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
So, the result is: −xlog(2)
The result is: −xlog(x5)−x5−xlog(2)
Method #2
-
Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=log(2x5) and let dv(x)=x21.
Then du(x)=x5.
To find v(x):
-
The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
Now evaluate the sub-integral.
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−x25)dx=−5∫x21dx
-
The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
So, the result is: x5
Method #3
-
Rewrite the integrand:
x2log(2x5)=x2log(x5)+x2log(2)
-
Integrate term-by-term:
-
Use integration by parts:
∫udv=uv−∫vdu
Let u(x)=log(x5) and let dv(x)=x21.
Then du(x)=x5.
To find v(x):
-
The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
Now evaluate the sub-integral.
-
The integral of a constant times a function is the constant times the integral of the function:
∫(−x25)dx=−5∫x21dx
-
The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
So, the result is: x5
-
The integral of a constant times a function is the constant times the integral of the function:
∫x2log(2)dx=log(2)∫x21dx
-
The integral of xn is n+1xn+1 when n=−1:
∫x21dx=−x1
So, the result is: −xlog(2)
The result is: −xlog(x5)−x5−xlog(2)
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Now simplify:
x−log(x5)−5−log(2)
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Add the constant of integration:
x−log(x5)−5−log(2)+constant
The answer is:
x−log(x5)−5−log(2)+constant
The answer (Indefinite)
[src]
/
|
| / 5\ / 5\
| log\2*x / 5 log(2) log\x /
| --------- dx = C - - - ------ - -------
| 2 x x x
| x
|
/
−xlog(2x5)−x5
The graph
5 log(64) log(486)
- + ------- - --------
6 2 3
5251(−32515log486+2565log64−325125+25625)
=
5 log(64) log(486)
- + ------- - --------
6 2 3
−3log(486)+65+2log(64)
Use the examples entering the upper and lower limits of integration.