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(2tan(x))÷(1-(tan(x))^(2))

Integral of (2tan(x))÷(1-(tan(x))^(2)) dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src]
  1               
  /               
 |                
 |    2*tan(x)    
 |  ----------- dx
 |         2      
 |  1 - tan (x)   
 |                
/                 
0                 
$$\int\limits_{0}^{1} \frac{2 \tan{\left(x \right)}}{1 - \tan^{2}{\left(x \right)}}\, dx$$
Integral(2*tan(x)/(1 - tan(x)^2), (x, 0, 1))
Detail solution
  1. The integral of a constant times a function is the constant times the integral of the function:

    1. There are multiple ways to do this integral.

      Method #1

      1. Let .

        Then let and substitute :

        1. The integral of a constant times a function is the constant times the integral of the function:

          1. Rewrite the integrand:

          2. The integral of a constant times a function is the constant times the integral of the function:

            1. Integrate term-by-term:

              1. The integral of is .

              1. The integral of a constant times a function is the constant times the integral of the function:

                1. The integral of is .

                So, the result is:

              The result is:

            So, the result is:

          So, the result is:

        Now substitute back in:

      Method #2

      1. Rewrite the integrand:

      2. The integral of a constant times a function is the constant times the integral of the function:

        1. Let .

          Then let and substitute :

          1. Rewrite the integrand:

          2. The integral of a constant times a function is the constant times the integral of the function:

            1. Integrate term-by-term:

              1. The integral of is .

              1. The integral of a constant times a function is the constant times the integral of the function:

                1. The integral of is .

                So, the result is:

              The result is:

            So, the result is:

          Now substitute back in:

        So, the result is:

    So, the result is:

  2. Now simplify:

  3. Add the constant of integration:


The answer is:

The answer (Indefinite) [src]
  /                                                         
 |                         /       2   \      /        2   \
 |   2*tan(x)           log\1 + tan (x)/   log\-1 + tan (x)/
 | ----------- dx = C + ---------------- - -----------------
 |        2                    2                   2        
 | 1 - tan (x)                                              
 |                                                          
/                                                           
$$-{{\log \left(2\,\sin ^2x-1\right)}\over{2}}$$
The graph
The answer [src]
nan
$$-{{\log \left(2\,\sin ^21-1\right)}\over{2}}$$
=
=
nan
$$\text{NaN}$$
Numerical answer [src]
-0.96117546625349
-0.96117546625349
The graph
Integral of (2tan(x))÷(1-(tan(x))^(2)) dx

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