Integral of tg(3x)dx dx

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 |  tan(3*x)*1 dx
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$$\int\limits_{0}^{1} \tan{\left(3 x \right)} 1\, dx$$

Integral(tan(3*x)*1, (x, 0, 1))

Detail solution

Let $u = 3 x$ .

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

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

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

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

The answer is:

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

The answer (Indefinite) [src]

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 |                     log(cos(3*x))
 | tan(3*x)*1 dx = C - -------------
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$${{\log \sec \left(3\,x\right)}\over{3}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$-{{\log \left(-\cos 3\right)}\over{3}}$$

nan

$$\text{NaN}$$

Упростить

Numerical answer [src]

-4.30726881219985

-4.30726881219985

The graph

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