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x/cos^2x
Solving integrals/ x/cos^2x

Integral of x/cos^2x dx

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$$\int\limits_{0}^{1} \frac{x}{\cos^{2}{\left(x \right)}}\, dx$$ 
Integral(x/cos(x)^2, (x, 0, 1))
The answer (Indefinite) [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|-| 
 |    x                \        \2//      \       \2//      \        \2//       \2/    \       \2//       \2/    \        \2//       \2/    \        \2//           \2/ 
 | ------- dx = C + ---------------- - --------------- - ---------------- + ----------------------- + ------------------------ - ------------------------ - ------------
 |    2                       2/x\               2/x\              2/x\                   2/x\                      2/x\                       2/x\                 2/x\
 | cos (x)            -1 + tan |-|       -1 + tan |-|      -1 + tan |-|           -1 + tan |-|              -1 + tan |-|               -1 + tan |-|         -1 + tan |-|
 |                             \2/                \2/               \2/                    \2/                       \2/                        \2/                  \2/
/
$$\int \frac{x}{\cos^{2}{\left(x \right)}}\, dx = C - \frac{2 x \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$
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The graph
Plot the graph f(x)
The answer [src] 
   /       2     \                                                                             2                                      2                             2         /       2     \
log\1 + tan (1/2)/          pi*I + log(1 - tan(1/2))   log(1 + tan(1/2))     2*tan(1/2)     tan (1/2)*(pi*I + log(1 - tan(1/2)))   tan (1/2)*log(1 + tan(1/2))   tan (1/2)*log\1 + tan (1/2)/
------------------ - pi*I - ------------------------ - ----------------- - -------------- + ------------------------------------ + --------------------------- - ----------------------------
          2                              2                       2                 2                           2                                  2                             2            
  -1 + tan (1/2)                 -1 + tan (1/2)          -1 + tan (1/2)    -1 + tan (1/2)              -1 + tan (1/2)                     -1 + tan (1/2)                -1 + tan (1/2)
$$\frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{2 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$ 
=
= 
   /       2     \                                                                             2                                      2                             2         /       2     \
log\1 + tan (1/2)/          pi*I + log(1 - tan(1/2))   log(1 + tan(1/2))     2*tan(1/2)     tan (1/2)*(pi*I + log(1 - tan(1/2)))   tan (1/2)*log(1 + tan(1/2))   tan (1/2)*log\1 + tan (1/2)/
------------------ - pi*I - ------------------------ - ----------------- - -------------- + ------------------------------------ + --------------------------- - ----------------------------
          2                              2                       2                 2                           2                                  2                             2            
  -1 + tan (1/2)                 -1 + tan (1/2)          -1 + tan (1/2)    -1 + tan (1/2)              -1 + tan (1/2)                     -1 + tan (1/2)                -1 + tan (1/2)
$$\frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{2 \tan{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$ 
log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2) - pi*i - (pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) - log(1 + tan(1/2))/(-1 + tan(1/2)^2) - 2*tan(1/2)/(-1 + tan(1/2)^2) + tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(-1 + tan(1/2)^2) + tan(1/2)^2*log(1 + tan(1/2))/(-1 + tan(1/2)^2) - tan(1/2)^2*log(1 + tan(1/2)^2)/(-1 + tan(1/2)^2)
Numerical answer [src] 
0.941781254268888
0.941781254268888
The graph
Integral of x/cos^2x dx

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

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