Integral of tan5x dx

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 |  tan(5*x) dx
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\int\limits_{0}^{1} \tan{\left(5 x \right)}\, dx

Integral(tan(5*x), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\tan{\left(5 x \right)} = \frac{\sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
There are multiple ways to do this integral.
Method #1
1. Let $u = \cos{\left(5 x \right)}$ .
  
  Then let $du = - 5 \sin{\left(5 x \right)} dx$ and substitute $- \frac{du}{5}$ :
  
  $\int \left(- \frac{1}{5 u}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{u}\, du = - \frac{\int \frac{1}{u}\, du}{5}$
    1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
    So, the result is: $- \frac{\log{\left(u \right)}}{5}$
  Now substitute $u$ back in:
  
  $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
Method #2
1. Let $u = 5 x$ .
  
  Then let $du = 5 dx$ and substitute $\frac{du}{5}$ :
  
  $\int \frac{\sin{\left(u \right)}}{5 \cos{\left(u \right)}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\sin{\left(u \right)}}{\cos{\left(u \right)}}\, du = \frac{\int \frac{\sin{\left(u \right)}}{\cos{\left(u \right)}}\, du}{5}$
    1. Let $u = \cos{\left(u \right)}$ .
      
      Then let $du = - \sin{\left(u \right)} du$ and substitute $- du$ :
      
      $\int \left(- \frac{1}{u}\right)\, du$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u}\, du = - \int \frac{1}{u}\, du$
      
      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: $- \frac{\log{\left(\cos{\left(u \right)} \right)}}{5}$
  Now substitute $u$ back in:
  
  $- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}$
Add the constant of integration:

$- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}+ \mathrm{constant}$

The answer is:

- \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}+ \mathrm{constant}

The answer (Indefinite) [src]

  /                               
 |                   log(cos(5*x))
 | tan(5*x) dx = C - -------------
 |                         5      
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\int \tan{\left(5 x \right)}\, dx = C - \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{5}

Simplify

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Numerical answer [src]

1.33441279257591

1.33441279257591

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Identical expressions

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Integral of tan5x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2