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sin(log(x))/

Integral of sin(log(x))/ dx

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The solution

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01sin(log(x))dx\int\limits_{0}^{1} \sin{\left(\log{\left(x \right)} \right)}\, dx
Integral(sin(log(x)), (x, 0, 1))
Detail solution
  1. Let u=log(x)u = \log{\left(x \right)}.

    Then let du=dxxdu = \frac{dx}{x} and substitute dudu:

    eusin(u)du\int e^{u} \sin{\left(u \right)}\, du

    1. Use integration by parts, noting that the integrand eventually repeats itself.

      1. For the integrand eusin(u)e^{u} \sin{\left(u \right)}:

        Let u(u)=sin(u)u{\left(u \right)} = \sin{\left(u \right)} and let dv(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

        Then eusin(u)du=eusin(u)eucos(u)du\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - \int e^{u} \cos{\left(u \right)}\, du.

      2. For the integrand eucos(u)e^{u} \cos{\left(u \right)}:

        Let u(u)=cos(u)u{\left(u \right)} = \cos{\left(u \right)} and let dv(u)=eu\operatorname{dv}{\left(u \right)} = e^{u}.

        Then eusin(u)du=eusin(u)eucos(u)+(eusin(u))du\int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)} + \int \left(- e^{u} \sin{\left(u \right)}\right)\, du.

      3. Notice that the integrand has repeated itself, so move it to one side:

        2eusin(u)du=eusin(u)eucos(u)2 \int e^{u} \sin{\left(u \right)}\, du = e^{u} \sin{\left(u \right)} - e^{u} \cos{\left(u \right)}

        Therefore,

        eusin(u)du=eusin(u)2eucos(u)2\int e^{u} \sin{\left(u \right)}\, du = \frac{e^{u} \sin{\left(u \right)}}{2} - \frac{e^{u} \cos{\left(u \right)}}{2}

    Now substitute uu back in:

    xsin(log(x))2xcos(log(x))2\frac{x \sin{\left(\log{\left(x \right)} \right)}}{2} - \frac{x \cos{\left(\log{\left(x \right)} \right)}}{2}

  2. Now simplify:

    2xcos(log(x)+π4)2- \frac{\sqrt{2} x \cos{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}

  3. Add the constant of integration:

    2xcos(log(x)+π4)2+constant- \frac{\sqrt{2} x \cos{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}


The answer is:

2xcos(log(x)+π4)2+constant- \frac{\sqrt{2} x \cos{\left(\log{\left(x \right)} + \frac{\pi}{4} \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src]
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 |                      x*sin(log(x))   x*cos(log(x))
 | sin(log(x)) dx = C + ------------- - -------------
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sin(log(x))dx=C+xsin(log(x))2xcos(log(x))2\int \sin{\left(\log{\left(x \right)} \right)}\, dx = C + \frac{x \sin{\left(\log{\left(x \right)} \right)}}{2} - \frac{x \cos{\left(\log{\left(x \right)} \right)}}{2}
The graph
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The answer [src]
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Numerical answer [src]
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The graph
Integral of sin(log(x))/ dx

    Use the examples entering the upper and lower limits of integration.