Integral of tan5x dx

The solution

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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

The graph

Plot the graph f(x)

Numerical answer [src]

1.33441279257591

1.33441279257591

The graph

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of tan5x dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2