Integral of (e*cos(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(x).
Then let du=xdx and substitute edu:
∫ecos(u)du
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The integral of a constant times a function is the constant times the integral of the function:
∫cos(u)du=e∫cos(u)du
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The integral of cosine is sine:
∫cos(u)du=sin(u)
So, the result is: esin(u)
Now substitute u back in:
esin(log(x))
Method #2
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Let u=x1.
Then let du=−x2dx and substitute −edu:
∫(−uecos(log(u1)))du
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The integral of a constant times a function is the constant times the integral of the function:
∫ucos(log(u1))du=−e∫ucos(log(u1))du
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Let u=u1.
Then let du=−u2du and substitute −du:
∫(−ucos(log(u)))du
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The integral of a constant times a function is the constant times the integral of the function:
∫ucos(log(u))du=−∫ucos(log(u))du
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Let u=log(u).
Then let du=udu and substitute du:
∫cos(u)du
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The integral of cosine is sine:
∫cos(u)du=sin(u)
Now substitute u back in:
sin(log(u))
So, the result is: −sin(log(u))
Now substitute u back in:
sin(log(u))
So, the result is: −esin(log(u))
Now substitute u back in:
esin(log(x))
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Add the constant of integration:
esin(log(x))+constant
The answer is:
esin(log(x))+constant
The answer (Indefinite)
[src]
/
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| E*cos(log(x))
| ------------- dx = C + E*sin(log(x))
| x
|
/
∫xecos(log(x))dx=C+esin(log(x))
−⟨−1,1⟩e
=
−⟨−1,1⟩e
Use the examples entering the upper and lower limits of integration.