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sinx/(cos^2)x-1

Integral of sinx/(cos^2)x+1 dx

The solution

You have entered [src]

  1                   
  /                   
 |                    
 |  / sin(x)      \   
 |  |-------*x + 1| dx
 |  |   2         |   
 |  \cos (x)      /   
 |                    
/                     
0

$$\int\limits_{0}^{1} \left(x \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)\, dx$$

Integral((sin(x)/cos(x)^2)*x + 1, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $- \frac{x \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{x}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
The result is: $x - \frac{x \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{x}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$
Now simplify:

$\frac{x \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)} + \frac{\left(\cos{\left(x \right)} - 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(- x + \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} - \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right)}{2} + \left(- x - \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} + \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}\right) \cos{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} - 1\right) \cos{\left(x \right)}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                                /       /x\\                     /        /x\\      2/x\    /        /x\\         2/x\        2/x\    /       /x\\
 |                              log|1 + tan|-||                  log|-1 + tan|-||   tan |-|*log|-1 + tan|-||    x*tan |-|     tan |-|*log|1 + tan|-||
 | / sin(x)      \                 \       \2//        x            \        \2//       \2/    \        \2//          \2/         \2/    \       \2//
 | |-------*x + 1| dx = C + x + --------------- - ------------ - ---------------- + ------------------------ - ------------ - -----------------------
 | |   2         |                        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 |-|     
 |                                         \2/             \2/              \2/                    \2/                  \2/                  \2/     
/

$$\int \left(x \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right)\, dx = C + x - \frac{x \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{x}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1} + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} - 1}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

                                                     2                                         2                                      2                       
          1          log(1 + tan(1/2))            tan (1/2)      pi*I + log(1 - tan(1/2))   tan (1/2)*(pi*I + log(1 - tan(1/2)))   tan (1/2)*log(1 + tan(1/2))
1 - -------------- + ----------------- - pi*I - -------------- - ------------------------ + ------------------------------------ - ---------------------------
            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)

$$\frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} + 1 - \frac{1}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - i \pi + \frac{\left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi}{-1 + \tan^{2}{\left(\frac{1}{2} \right)}}$$

                                                     2                                         2                                      2                       
          1          log(1 + tan(1/2))            tan (1/2)      pi*I + log(1 - tan(1/2))   tan (1/2)*(pi*I + log(1 - tan(1/2)))   tan (1/2)*log(1 + tan(1/2))
1 - -------------- + ----------------- - pi*I - -------------- - ------------------------ + ------------------------------------ - ---------------------------
            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 - 1/(-1 + tan(1/2)^2) + log(1 + tan(1/2))/(-1 + tan(1/2)^2) - pi*i - tan(1/2)^2/(-1 + tan(1/2)^2) - (pi*i + log(1 - 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)

Numerical answer [src]

1.62462454679741

1.62462454679741

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

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Identical expressions

Similar expressions

Integral of sinx/(cos^2)x+1 dx

Limits of integration:

The graph:

Piecewise:

The solution