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(1+tgx)/(1-tgx)
Solving integrals/ (1+tgx)/(1-tgx)

Integral of (1+tgx)/(1-tgx) dx

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  1              
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 |  1 + tan(x)   
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 |  1 - tan(x)   
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0
∫01tan⁡(x)+1−tan⁡(x)+1 dx\int\limits_{0}^{1} \frac{\tan{\left(x \right)} + 1}{- \tan{\left(x \right)} + 1}\, dx
Simplify
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      tan⁡(x)+11−tan⁡(x)=−tan⁡(x)−1tan⁡(x)−1\frac{\tan{\left(x \right)} + 1}{1 - \tan{\left(x \right)}} = \frac{- \tan{\left(x \right)} - 1}{\tan{\left(x \right)} - 1}

    2. Rewrite the integrand:

      −tan⁡(x)−1tan⁡(x)−1=−tan⁡(x)tan⁡(x)−1−1tan⁡(x)−1\frac{- \tan{\left(x \right)} - 1}{\tan{\left(x \right)} - 1} = - \frac{\tan{\left(x \right)}}{\tan{\left(x \right)} - 1} - \frac{1}{\tan{\left(x \right)} - 1}

    3. Integrate term-by-term:

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

        ∫(−tan⁡(x)tan⁡(x)−1) dx=−∫tan⁡(x)tan⁡(x)−1 dx\int \left(- \frac{\tan{\left(x \right)}}{\tan{\left(x \right)} - 1}\right)\, dx = - \int \frac{\tan{\left(x \right)}}{\tan{\left(x \right)} - 1}\, dx

        1. Don't know the steps in finding this integral.

          But the integral is

          x2+log⁡(tan⁡(x)−1)2−log⁡(tan⁡2(x)+1)4\frac{x}{2} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}

        So, the result is: −x2−log⁡(tan⁡(x)−1)2+log⁡(tan⁡2(x)+1)4- \frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}

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

        ∫(−1tan⁡(x)−1) dx=−∫1tan⁡(x)−1 dx\int \left(- \frac{1}{\tan{\left(x \right)} - 1}\right)\, dx = - \int \frac{1}{\tan{\left(x \right)} - 1}\, dx

        1. Don't know the steps in finding this integral.

          But the integral is

          −x2+log⁡(tan⁡(x)−1)2−log⁡(tan⁡2(x)+1)4- \frac{x}{2} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}

        So, the result is: x2−log⁡(tan⁡(x)−1)2+log⁡(tan⁡2(x)+1)4\frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}

      The result is: −log⁡(tan⁡(x)−1)+log⁡(tan⁡2(x)+1)2- \log{\left(\tan{\left(x \right)} - 1 \right)} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2}

    Method #2

    1. Rewrite the integrand:

      tan⁡(x)+11−tan⁡(x)=tan⁡(x)1−tan⁡(x)+11−tan⁡(x)\frac{\tan{\left(x \right)} + 1}{1 - \tan{\left(x \right)}} = \frac{\tan{\left(x \right)}}{1 - \tan{\left(x \right)}} + \frac{1}{1 - \tan{\left(x \right)}}

    2. Integrate term-by-term:

      1. Rewrite the integrand:

        tan⁡(x)1−tan⁡(x)=−tan⁡(x)tan⁡(x)−1\frac{\tan{\left(x \right)}}{1 - \tan{\left(x \right)}} = - \frac{\tan{\left(x \right)}}{\tan{\left(x \right)} - 1}

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

        ∫(−tan⁡(x)tan⁡(x)−1) dx=−∫tan⁡(x)tan⁡(x)−1 dx\int \left(- \frac{\tan{\left(x \right)}}{\tan{\left(x \right)} - 1}\right)\, dx = - \int \frac{\tan{\left(x \right)}}{\tan{\left(x \right)} - 1}\, dx

        1. Don't know the steps in finding this integral.

          But the integral is

          x2+log⁡(tan⁡(x)−1)2−log⁡(tan⁡2(x)+1)4\frac{x}{2} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}

        So, the result is: −x2−log⁡(tan⁡(x)−1)2+log⁡(tan⁡2(x)+1)4- \frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}

      1. Rewrite the integrand:

        11−tan⁡(x)=−1tan⁡(x)−1\frac{1}{1 - \tan{\left(x \right)}} = - \frac{1}{\tan{\left(x \right)} - 1}

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

        ∫(−1tan⁡(x)−1) dx=−∫1tan⁡(x)−1 dx\int \left(- \frac{1}{\tan{\left(x \right)} - 1}\right)\, dx = - \int \frac{1}{\tan{\left(x \right)} - 1}\, dx

        1. Don't know the steps in finding this integral.

          But the integral is

          −x2+log⁡(tan⁡(x)−1)2−log⁡(tan⁡2(x)+1)4- \frac{x}{2} + \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{2} - \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}

        So, the result is: x2−log⁡(tan⁡(x)−1)2+log⁡(tan⁡2(x)+1)4\frac{x}{2} - \frac{\log{\left(\tan{\left(x \right)} - 1 \right)}}{2} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{4}

      The result is: −log⁡(tan⁡(x)−1)+log⁡(tan⁡2(x)+1)2- \log{\left(\tan{\left(x \right)} - 1 \right)} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2}

  2. Now simplify:

    −log⁡(tan⁡(x)−1)+log⁡(1cos⁡2(x))2- \log{\left(\tan{\left(x \right)} - 1 \right)} + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{2}

  3. Add the constant of integration:

    −log⁡(tan⁡(x)−1)+log⁡(1cos⁡2(x))2+constant- \log{\left(\tan{\left(x \right)} - 1 \right)} + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{2}+ \mathrm{constant}

The answer is:

−log⁡(tan⁡(x)−1)+log⁡(1cos⁡2(x))2+constant- \log{\left(\tan{\left(x \right)} - 1 \right)} + \frac{\log{\left(\frac{1}{\cos^{2}{\left(x \right)}} \right)}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                       
 |                        /       2   \                   
 | 1 + tan(x)          log\1 + tan (x)/                   
 | ---------- dx = C + ---------------- - log(-1 + tan(x))
 | 1 - tan(x)                 2                           
 |                                                        
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log⁡(tan⁡2x+1)2−log⁡(tan⁡x−1){{\log \left(\tan ^2x+1\right)}\over{2}}-\log \left(\tan x-1\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-5000050000
Plot the graph f(x)
The answer [src] 
nan
log⁡(tan⁡21+1)2−log⁡(tan⁡1−1){{\log \left(\tan ^21+1\right)}\over{2}}-\log \left(\tan 1-1\right) 
=
= 
nan
NaN\text{NaN}
Simplify
Numerical answer [src] 
-1.5989832649916
-1.5989832649916
The graph
Integral of (1+tgx)/(1-tgx) dx

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

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