Integral of sin(log(x))/ dx
The solution
Detail solution
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Let u=log(x).
Then let du=xdx and substitute du:
∫eusin(u)du
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Use integration by parts, noting that the integrand eventually repeats itself.
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For the integrand eusin(u):
Let u(u)=sin(u) and let dv(u)=eu.
Then ∫eusin(u)du=eusin(u)−∫eucos(u)du.
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For the integrand eucos(u):
Let u(u)=cos(u) and let dv(u)=eu.
Then ∫eusin(u)du=eusin(u)−eucos(u)+∫(−eusin(u))du.
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Notice that the integrand has repeated itself, so move it to one side:
2∫eusin(u)du=eusin(u)−eucos(u)
Therefore,
∫eusin(u)du=2eusin(u)−2eucos(u)
Now substitute u back in:
2xsin(log(x))−2xcos(log(x))
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Now simplify:
−22xcos(log(x)+4π)
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Add the constant of integration:
−22xcos(log(x)+4π)+constant
The answer is:
−22xcos(log(x)+4π)+constant
The answer (Indefinite)
[src]
/
| x*sin(log(x)) x*cos(log(x))
| sin(log(x)) dx = C + ------------- - -------------
| 2 2
/
∫sin(log(x))dx=C+2xsin(log(x))−2xcos(log(x))
The graph
Use the examples entering the upper and lower limits of integration.