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Resolución de integrales/ sec²×tan×dx

Integral de sec²×tan×dx dx

Límites de integración:

interior superior
v

Gráfico:

interior superior

Definida a trozos:

Solución

Ha introducido [src] 
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∫01xtan⁡(d)sec⁡2(x) dx\int\limits_{0}^{1} x \tan{\left(d \right)} \sec^{2}{\left(x \right)}\, dx 
Integral((sec(x)^2*tan(d))*x, (x, 0, 1))
Solución detallada

  1. Vuelva a escribir el integrando:

    xtan⁡(d)sec⁡2(x)=xtan⁡(d)cos⁡2(x)x \tan{\left(d \right)} \sec^{2}{\left(x \right)} = \frac{x \tan{\left(d \right)}}{\cos^{2}{\left(x \right)}}

  2. La integral del producto de una función por una constante es la constante por la integral de esta función:

    ∫xtan⁡(d)cos⁡2(x) dx=tan⁡(d)∫xcos⁡2(x) dx\int \frac{x \tan{\left(d \right)}}{\cos^{2}{\left(x \right)}}\, dx = \tan{\left(d \right)} \int \frac{x}{\cos^{2}{\left(x \right)}}\, dx

    1. No puedo encontrar los pasos en la búsqueda de esta integral.

      Pero la integral

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

    Por lo tanto, el resultado es: (−2xtan⁡(x2)tan⁡2(x2)−1+log⁡(tan⁡(x2)−1)tan⁡2(x2)tan⁡2(x2)−1−log⁡(tan⁡(x2)−1)tan⁡2(x2)−1+log⁡(tan⁡(x2)+1)tan⁡2(x2)tan⁡2(x2)−1−log⁡(tan⁡(x2)+1)tan⁡2(x2)−1−log⁡(tan⁡2(x2)+1)tan⁡2(x2)tan⁡2(x2)−1+log⁡(tan⁡2(x2)+1)tan⁡2(x2)−1)tan⁡(d)\left(- \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}\right) \tan{\left(d \right)}

  3. Ahora simplificar:

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

  4. Añadimos la constante de integración:

    (2x(tan⁡2(x2)−1)sin⁡(x)+(log⁡(tan⁡(x2)−1)+log⁡(tan⁡(x2)+1))(cos⁡(x)−1)(tan⁡2(x2)−1)+2(−log⁡(2cos⁡(x)+1)tan⁡2(x2)+log⁡(2cos⁡(x)+1)−log⁡(tan⁡(x2)−1)−log⁡(tan⁡(x2)+1))cos⁡(x))tan⁡(d)2(tan⁡2(x2)−1)cos⁡(x)+constant\frac{\left(2 x \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} + \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) + 2 \left(- \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}\right) \tan{\left(d \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}+ \mathrm{constant}

Respuesta:

(2x(tan⁡2(x2)−1)sin⁡(x)+(log⁡(tan⁡(x2)−1)+log⁡(tan⁡(x2)+1))(cos⁡(x)−1)(tan⁡2(x2)−1)+2(−log⁡(2cos⁡(x)+1)tan⁡2(x2)+log⁡(2cos⁡(x)+1)−log⁡(tan⁡(x2)−1)−log⁡(tan⁡(x2)+1))cos⁡(x))tan⁡(d)2(tan⁡2(x2)−1)cos⁡(x)+constant\frac{\left(2 x \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \sin{\left(x \right)} + \left(\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) + 2 \left(- \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} \tan^{2}{\left(\frac{x}{2} \right)} + \log{\left(\frac{2}{\cos{\left(x \right)} + 1} \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}\right) \tan{\left(d \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}+ \mathrm{constant}

Respuesta (Indefinida) [src] 
  /                          /   /       2/x\\      /       /x\\      /        /x\\      2/x\    /       /x\\      2/x\    /        /x\\      2/x\    /       2/x\\           /x\ \       
 |                           |log|1 + tan |-||   log|1 + tan|-||   log|-1 + tan|-||   tan |-|*log|1 + tan|-||   tan |-|*log|-1 + tan|-||   tan |-|*log|1 + tan |-||    2*x*tan|-| |       
 |    2                      |   \        \2//      \       \2//      \        \2//       \2/    \       \2//       \2/    \        \2//       \2/    \        \2//           \2/ |       
 | sec (x)*tan(d)*x dx = C + |---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------|*tan(d)
 |                           |          2/x\               2/x\              2/x\                   2/x\                      2/x\                       2/x\                 2/x\|       
/                            |  -1 + tan |-|       -1 + tan |-|      -1 + tan |-|           -1 + tan |-|              -1 + tan |-|               -1 + tan |-|         -1 + tan |-||       
                             \           \2/                \2/               \2/                    \2/                       \2/                        \2/                  \2//
∫xtan⁡(d)sec⁡2(x) dx=C+(−2xtan⁡(x2)tan⁡2(x2)−1+log⁡(tan⁡(x2)−1)tan⁡2(x2)tan⁡2(x2)−1−log⁡(tan⁡(x2)−1)tan⁡2(x2)−1+log⁡(tan⁡(x2)+1)tan⁡2(x2)tan⁡2(x2)−1−log⁡(tan⁡(x2)+1)tan⁡2(x2)−1−log⁡(tan⁡2(x2)+1)tan⁡2(x2)tan⁡2(x2)−1+log⁡(tan⁡2(x2)+1)tan⁡2(x2)−1)tan⁡(d)\int x \tan{\left(d \right)} \sec^{2}{\left(x \right)}\, dx = C + \left(- \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}\right) \tan{\left(d \right)}
Simplificar
Respuesta [src] 
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tan⁡(d)∫01xsec⁡2(x) dx\tan{\left(d \right)} \int\limits_{0}^{1} x \sec^{2}{\left(x \right)}\, dx 
=
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tan⁡(d)∫01xsec⁡2(x) dx\tan{\left(d \right)} \int\limits_{0}^{1} x \sec^{2}{\left(x \right)}\, dx 
Integral(x*sec(x)^2, (x, 0, 1))*tan(d)

    Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.

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