Integral of tg4xdx dx

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 |  tan(4*x)*1 dx
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$$\int\limits_{0}^{1} \tan{\left(4 x \right)} 1\, dx$$

Simplify

Detail solution

Let $u = 4 x$ .

Then let $du = 4 dx$ and substitute $\frac{du}{4}$ :

$\int \frac{\tan{\left(u \right)}}{16}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{\tan{\left(u \right)}}{4}\, du = \frac{\int \tan{\left(u \right)}\, du}{4}$
  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: $- \frac{\log{\left(\cos{\left(u \right)} \right)}}{4}$
Now substitute $u$ back in:

$- \frac{\log{\left(\cos{\left(4 x \right)} \right)}}{4}$
Add the constant of integration:

$- \frac{\log{\left(\cos{\left(4 x \right)} \right)}}{4}+ \mathrm{constant}$

The answer is:

$- \frac{\log{\left(\cos{\left(4 x \right)} \right)}}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                 
 |                     log(cos(4*x))
 | tan(4*x)*1 dx = C - -------------
 |                           4      
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$${{\log \sec \left(4\,x\right)}\over{4}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$-{{\log \left(-\cos 4\right)}\over{4}}$$

nan

$$\text{NaN}$$

Simplify

Numerical answer [src]

1.6900933662682

1.6900933662682

The graph

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