Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of 1/(2+x^2)
Integral of x/(x^3-3x+2)
Integral of (x-2)^2dx
Integral of 8/x^2
Identical expressions
tan^4x/(one -tan^4x)
tangent of to the power of 4x divide by (1 minus tangent of to the power of 4x)
tangent of to the power of 4x divide by (one minus tangent of to the power of 4x)
tan4x/(1-tan4x)
tan4x/1-tan4x
tan⁴x/(1-tan⁴x)
tan^4x/1-tan^4x
tan^4x divide by (1-tan^4x)
tan^4x/(1-tan^4x)dx
Similar expressions
tan^4x/(1+tan^4x)

Solving integrals/ tan^4x/(1-tan^4x)

Integral of tan^4x/(1-tan^4x) dx

The solution

You have entered [src]

  1               
  /               
 |                
 |       4        
 |    tan (x)     
 |  ----------- dx
 |         4      
 |  1 - tan (x)   
 |                
/                 
0

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

Integral(tan(x)^4/(1 - tan(x)^4), (x, 0, 1))

Detail solution

Rewrite the integrand:

$\frac{\tan^{4}{\left(x \right)}}{1 - \tan^{4}{\left(x \right)}} = - \frac{\tan^{4}{\left(x \right)}}{\tan^{4}{\left(x \right)} - 1}$
The integral of a constant times a function is the constant times the integral of the function:

$\int \left(- \frac{\tan^{4}{\left(x \right)}}{\tan^{4}{\left(x \right)} - 1}\right)\, dx = - \int \frac{\tan^{4}{\left(x \right)}}{\tan^{4}{\left(x \right)} - 1}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{4 x \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{4 x}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{2 \tan{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8}$
So, the result is: $- \frac{4 x \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{4 x}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{2 \tan{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8}$
Now simplify:

$- \frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8} + \frac{\sin{\left(2 x \right)}}{8}$
Add the constant of integration:

$- \frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8} + \frac{\sin{\left(2 x \right)}}{8}+ \mathrm{constant}$

The answer is:

$- \frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8} + \frac{\sin{\left(2 x \right)}}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                                            
 |                                                                                                                                                             
 |      4                                                                                       2                         2                               2    
 |   tan (x)            log(1 + tan(x))   log(-1 + tan(x))        4*x           2*tan(x)     tan (x)*log(1 + tan(x))   tan (x)*log(-1 + tan(x))    4*x*tan (x) 
 | ----------- dx = C + --------------- - ---------------- - ------------- + ------------- + ----------------------- - ------------------------ - -------------
 |        4                       2                 2                 2               2                    2                         2                     2   
 | 1 - tan (x)           8 + 8*tan (x)     8 + 8*tan (x)     8 + 8*tan (x)   8 + 8*tan (x)        8 + 8*tan (x)             8 + 8*tan (x)         8 + 8*tan (x)
 |                                                                                                                                                             
/

$$\int \frac{\tan^{4}{\left(x \right)}}{1 - \tan^{4}{\left(x \right)}}\, dx = C - \frac{4 x \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{4 x}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)} \tan^{2}{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{\log{\left(\tan{\left(x \right)} + 1 \right)}}{8 \tan^{2}{\left(x \right)} + 8} + \frac{2 \tan{\left(x \right)}}{8 \tan^{2}{\left(x \right)} + 8}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

nan

$$\text{NaN}$$

nan

$$\text{NaN}$$

nan

Numerical answer [src]

-0.545789771331317

-0.545789771331317

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 tan^4x/(1-tan^4x) dx

Limits of integration:

The graph:

Piecewise:

The solution