Integral of sin(ln(x))/sqrt(x) dx
The solution
Detail solution
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Let u=log(x).
Then let du=xdx and substitute du:
∫e2usin(u)du
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Use integration by parts, noting that the integrand eventually repeats itself.
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For the integrand e2usin(u):
Let u(u)=sin(u) and let dv(u)=e2u.
Then ∫e2usin(u)du=2e2usin(u)−∫2e2ucos(u)du.
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For the integrand 2e2ucos(u):
Let u(u)=2cos(u) and let dv(u)=e2u.
Then ∫e2usin(u)du=2e2usin(u)−4e2ucos(u)+∫(−4e2usin(u))du.
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Notice that the integrand has repeated itself, so move it to one side:
5∫e2usin(u)du=2e2usin(u)−4e2ucos(u)
Therefore,
∫e2usin(u)du=52e2usin(u)−54e2ucos(u)
Now substitute u back in:
52xsin(log(x))−54xcos(log(x))
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Now simplify:
52x(sin(log(x))−2cos(log(x)))
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Add the constant of integration:
52x(sin(log(x))−2cos(log(x)))+constant
The answer is:
52x(sin(log(x))−2cos(log(x)))+constant
The answer (Indefinite)
[src]
/
| ___ ___
| sin(log(x)) 4*\/ x *cos(log(x)) 2*\/ x *sin(log(x))
| ----------- dx = C - ------------------- + -------------------
| ___ 5 5
| \/ x
|
/
∫xsin(log(x))dx=C+52xsin(log(x))−54xcos(log(x))
⟨−∞,∞⟩
=
⟨−∞,∞⟩
Use the examples entering the upper and lower limits of integration.