Integral of 12ctg3x dx

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 |  12*cot(3*x) dx
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16

$$\int\limits_{\frac{\pi}{16}}^{\frac{\pi}{6}} 12 \cot{\left(3 x \right)}\, dx$$

Integral(12*cot(3*x), (x, pi/16, pi/6))

Detail solution

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

$\int 12 \cot{\left(3 x \right)}\, dx = 12 \int \cot{\left(3 x \right)}\, dx$
1. Rewrite the integrand:
  
  $\cot{\left(3 x \right)} = \frac{\cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}$
2. There are multiple ways to do this integral.
  Method #1
  1. Let $u = \sin{\left(3 x \right)}$ .
    
    Then let $du = 3 \cos{\left(3 x \right)} dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{1}{3 u}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      So, the result is: $\frac{\log{\left(u \right)}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{3}$
  Method #2
  1. Let $u = 3 x$ .
    
    Then let $du = 3 dx$ and substitute $\frac{du}{3}$ :
    
    $\int \frac{\cos{\left(u \right)}}{3 \sin{\left(u \right)}}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}\, du = \frac{\int \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}\, du}{3}$
      
      Let $u = \sin{\left(u \right)}$ .
      
      Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(\sin{\left(u \right)} \right)}$
      
      So, the result is: $\frac{\log{\left(\sin{\left(u \right)} \right)}}{3}$
    Now substitute $u$ back in:
    
    $\frac{\log{\left(\sin{\left(3 x \right)} \right)}}{3}$
So, the result is: $4 \log{\left(\sin{\left(3 x \right)} \right)}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                    
 |                                     
 | 12*cot(3*x) dx = C + 4*log(sin(3*x))
 |                                     
/

$$\int 12 \cot{\left(3 x \right)}\, dx = C + 4 \log{\left(\sin{\left(3 x \right)} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      /   /3*pi\\
-4*log|sin|----||
      \   \ 16 //

$$- 4 \log{\left(\sin{\left(\frac{3 \pi}{16} \right)} \right)}$$

=

      /   /3*pi\\
-4*log|sin|----||
      \   \ 16 //

$$- 4 \log{\left(\sin{\left(\frac{3 \pi}{16} \right)} \right)}$$

-4*log(sin(3*pi/16))

Numerical answer [src]

2.35104098326329

2.35104098326329

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Integral of 12ctg3x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2