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Solving integrals/ log(sin(x)^3)

Integral of log(sin(x)^3) dx

Limits of integration:

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The solution

You have entered [src] 
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∫01log⁡(sin⁡3(x)) dx\int\limits_{0}^{1} \log{\left(\sin^{3}{\left(x \right)} \right)}\, dx 
Integral(log(sin(x)^3), (x, 0, 1))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=log⁡(sin⁡3(x))u{\left(x \right)} = \log{\left(\sin^{3}{\left(x \right)} \right)} and let dv⁡(x)=1\operatorname{dv}{\left(x \right)} = 1.

    Then du⁡(x)=3cos⁡(x)sin⁡(x)\operatorname{du}{\left(x \right)} = \frac{3 \cos{\left(x \right)}}{\sin{\left(x \right)}}.

    To find v(x)v{\left(x \right)}:

    1. The integral of a constant is the constant times the variable of integration:

      ∫1 dx=x\int 1\, dx = x

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫3xcos⁡(x)sin⁡(x) dx=3∫xcos⁡(x)sin⁡(x) dx\int \frac{3 x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx = 3 \int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx

    1. Don't know the steps in finding this integral.

      But the integral is

      ∫xcos⁡(x)sin⁡(x) dx\int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx

    So, the result is: 3∫xcos⁡(x)sin⁡(x) dx3 \int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx

  3. Now simplify:

    xlog⁡(sin⁡3(x))−3∫xtan⁡(x) dxx \log{\left(\sin^{3}{\left(x \right)} \right)} - 3 \int \frac{x}{\tan{\left(x \right)}}\, dx

  4. Add the constant of integration:

    xlog⁡(sin⁡3(x))−3∫xtan⁡(x) dx+constantx \log{\left(\sin^{3}{\left(x \right)} \right)} - 3 \int \frac{x}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}

The answer is:

xlog⁡(sin⁡3(x))−3∫xtan⁡(x) dx+constantx \log{\left(\sin^{3}{\left(x \right)} \right)} - 3 \int \frac{x}{\tan{\left(x \right)}}\, dx+ \mathrm{constant}

The answer (Indefinite) [src] 
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∫log⁡(sin⁡3(x)) dx=C+xlog⁡(sin⁡3(x))−3∫xcos⁡(x)sin⁡(x) dx\int \log{\left(\sin^{3}{\left(x \right)} \right)}\, dx = C + x \log{\left(\sin^{3}{\left(x \right)} \right)} - 3 \int \frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}}\, dx
Simplify
The answer [src] 
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∫01log⁡(sin⁡3(x)) dx\int\limits_{0}^{1} \log{\left(\sin^{3}{\left(x \right)} \right)}\, dx 
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∫01log⁡(sin⁡3(x)) dx\int\limits_{0}^{1} \log{\left(\sin^{3}{\left(x \right)} \right)}\, dx 
Integral(log(sin(x)^3), (x, 0, 1))
Numerical answer [src] 
-3.17016061797475
-3.17016061797475

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

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