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

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

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∫01(tan⁡(x)+1)11−tan⁡(x)1 dx\int\limits_{0}^{1} \left(\tan{\left(x \right)} + 1\right) \frac{1}{1 - \tan{\left(x \right)}} 1\, dx 
Integral((1 + tan(x))*1/(1 - tan(x)), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Rewrite the integrand:

      (tan⁡(x)+1)11−tan⁡(x)1=−tan⁡(x)−1tan⁡(x)−1\left(\tan{\left(x \right)} + 1\right) \frac{1}{1 - \tan{\left(x \right)}} 1 = \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)+1)11−tan⁡(x)1=tan⁡(x)1−tan⁡(x)+11−tan⁡(x)\left(\tan{\left(x \right)} + 1\right) \frac{1}{1 - \tan{\left(x \right)}} 1 = \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                 log\1 + tan (x)/                   
 | (1 + tan(x))*----------*1 dx = C + ---------------- - log(-1 + tan(x))
 |              1 - tan(x)                   2                           
 |                                                                       
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∫(tan⁡(x)+1)11−tan⁡(x)1 dx=C−log⁡(tan⁡(x)−1)+log⁡(tan⁡2(x)+1)2\int \left(\tan{\left(x \right)} + 1\right) \frac{1}{1 - \tan{\left(x \right)}} 1\, dx = C - \log{\left(\tan{\left(x \right)} - 1 \right)} + \frac{\log{\left(\tan^{2}{\left(x \right)} + 1 \right)}}{2}
Simplify
The graph
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The answer [src] 
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Numerical answer [src] 
-1.5989832649916
-1.5989832649916
The graph
Integral of ((1+tgx)/(1-tgx))*dx dx

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

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