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Solving integrals/ 12*ctg(3*x)

Integral of 12*ctg(3*x) dx

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The solution

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∫π12π612cot⁡(3x) dx\int\limits_{\frac{\pi}{12}}^{\frac{\pi}{6}} 12 \cot{\left(3 x \right)}\, dx 
Integral(12*cot(3*x), (x, pi/12, pi/6))
Detail solution

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

    ∫12cot⁡(3x) dx=12∫cot⁡(3x) dx\int 12 \cot{\left(3 x \right)}\, dx = 12 \int \cot{\left(3 x \right)}\, dx

    1. Rewrite the integrand:

      cot⁡(3x)=cos⁡(3x)sin⁡(3x)\cot{\left(3 x \right)} = \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}

    2. There are multiple ways to do this integral.

      Method #1

      1. Let u=sin⁡(3x)u = \sin{\left(3 x \right)}.

        Then let du=3cos⁡(3x)dxdu = 3 \cos{\left(3 x \right)} dx and substitute du3\frac{du}{3}:

        ∫13u du\int \frac{1}{3 u}\, du

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

          ∫1u du=∫1u du3\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}

          1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

          So, the result is: log⁡(u)3\frac{\log{\left(u \right)}}{3}

        Now substitute uu back in:

        log⁡(sin⁡(3x))3\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{3}

      Method #2

      1. Let u=3xu = 3 x.

        Then let du=3dxdu = 3 dx and substitute du3\frac{du}{3}:

        ∫cos⁡(u)3sin⁡(u) du\int \frac{\cos{\left(u \right)}}{3 \sin{\left(u \right)}}\, du

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

          ∫cos⁡(u)sin⁡(u) du=∫cos⁡(u)sin⁡(u) du3\int \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}\, du = \frac{\int \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}\, du}{3}

          1. Let u=sin⁡(u)u = \sin{\left(u \right)}.

            Then let du=cos⁡(u)dudu = \cos{\left(u \right)} du and substitute dudu:

            ∫1u du\int \frac{1}{u}\, du

            1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

            Now substitute uu back in:

            log⁡(sin⁡(u))\log{\left(\sin{\left(u \right)} \right)}

          So, the result is: log⁡(sin⁡(u))3\frac{\log{\left(\sin{\left(u \right)} \right)}}{3}

        Now substitute uu back in:

        log⁡(sin⁡(3x))3\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{3}

    So, the result is: 4log⁡(sin⁡(3x))4 \log{\left(\sin{\left(3 x \right)} \right)}

  2. Add the constant of integration:

    4log⁡(sin⁡(3x))+constant4 \log{\left(\sin{\left(3 x \right)} \right)}+ \mathrm{constant}

The answer is:

4log⁡(sin⁡(3x))+constant4 \log{\left(\sin{\left(3 x \right)} \right)}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                    
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 | 12*cot(3*x) dx = C + 4*log(sin(3*x))
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∫12cot⁡(3x) dx=C+4log⁡(sin⁡(3x))\int 12 \cot{\left(3 x \right)}\, dx = C + 4 \log{\left(\sin{\left(3 x \right)} \right)}
Simplify
The graph
0.2750.3000.3250.3500.3750.4000.4250.4500.4750.50020-10
Plot the graph f(x)
The answer [src] 
      /  ___\
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-4*log|-----|
      \  2  /
−4log⁡(22)- 4 \log{\left(\frac{\sqrt{2}}{2} \right)} 
=
= 
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-4*log|-----|
      \  2  /
−4log⁡(22)- 4 \log{\left(\frac{\sqrt{2}}{2} \right)} 
-4*log(sqrt(2)/2)
Numerical answer [src] 
1.38629436111989
1.38629436111989

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

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