Integral of ctg(3x-2)dx dx

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 |  cot(3*x - 2)*1 dx
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0

$$\int\limits_{0}^{1} \cot{\left(3 x - 2 \right)} 1\, dx$$

Integral(cot(3*x - 1*2)*1, (x, 0, 1))

Detail solution

Let $u = 3 x - 2$ .

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

$\int \frac{\cot{\left(u \right)}}{9}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\cot{\left(u \right)}}{3}\, du = \frac{\int \cot{\left(u \right)}\, du}{3}$
  1. Rewrite the integrand:
    
    $\cot{\left(u \right)} = \frac{\cos{\left(u \right)}}{\sin{\left(u \right)}}$
  2. Let $u = \sin{\left(u \right)}$ .
    
    Then let $du = \cos{\left(u \right)} du$ and substitute $du$ :
    
    $\int \frac{1}{u}\, du$
    1. 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:

$\frac{\log{\left(\sin{\left(3 x - 2 \right)} \right)}}{3}$
Now simplify:

$\frac{\log{\left(\sin{\left(3 x - 2 \right)} \right)}}{3}$
Add the constant of integration:

$\frac{\log{\left(\sin{\left(3 x - 2 \right)} \right)}}{3}+ \mathrm{constant}$

The answer is:

$\frac{\log{\left(\sin{\left(3 x - 2 \right)} \right)}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                         
 |                         log(sin(3*x - 2))
 | cot(3*x - 2)*1 dx = C + -----------------
 |                                 3        
/

$${{\log \sin \left(3\,x-2\right)}\over{3}}$$

Simplify

The answer [src]

                    /       2   \                    /       2   \
  log(-tan(2))   log\1 + tan (1)/   log(tan(1))   log\1 + tan (2)/
- ------------ - ---------------- + ----------- + ----------------
       3                6                3               6

$${{\log \sin 1}\over{3}}-{{\log \sin 2}\over{3}}$$

=

                    /       2   \                    /       2   \
  log(-tan(2))   log\1 + tan (1)/   log(tan(1))   log\1 + tan (2)/
- ------------ - ---------------- + ----------- + ----------------
       3                6                3               6

$$- \frac{\log{\left(- \tan{\left(2 \right)} \right)}}{3} - \frac{\log{\left(1 + \tan^{2}{\left(1 \right)} \right)}}{6} + \frac{\log{\left(\tan{\left(1 \right)} \right)}}{3} + \frac{\log{\left(1 + \tan^{2}{\left(2 \right)} \right)}}{6}$$

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Numerical answer [src]

-38.9947548058377

-38.9947548058377

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Integral of ctg(3x-2)dx dx

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