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x/(cosh(x))^2

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  • Integral of d{x}:
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  • Identical expressions

  • x/(cosh(x))^ two
  • x divide by ( hyperbolic co sinus of e of ine of (x)) squared
  • x divide by ( hyperbolic co sinus of e of ine of (x)) to the power of two
  • x/(cosh(x))2
  • x/coshx2
  • x/(cosh(x))²
  • x/(cosh(x)) to the power of 2
  • x/coshx^2
  • x divide by (cosh(x))^2
  • x/(cosh(x))^2dx
Solving integrals/ x/(cosh(x))^2

Integral of x/(cosh(x))^2 dx

Limits of integration:

from to
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The graph:

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Piecewise:

The solution

You have entered [src] 
  1            
  /            
 |             
 |     x       
 |  -------- dx
 |      2      
 |  cosh (x)   
 |             
/              
0
$$\int\limits_{0}^{1} \frac{x}{\cosh^{2}{\left(x \right)}}\, dx$$ 
Integral(x/(cosh(x)^2), (x, 0, 1))
The answer (Indefinite) [src] 
  /                                    /        2/x\\        /        /x\\          2/x\        2/x\    /        2/x\\           /x\          2/x\    /        /x\\
 |                                  log|1 + tanh |-||   2*log|1 + tanh|-||    x*tanh |-|    tanh |-|*log|1 + tanh |-||   2*x*tanh|-|    2*tanh |-|*log|1 + tanh|-||
 |    x                   x            \         \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\       
 | cosh (x)          1 + tanh |-|      1 + tanh |-|        1 + tanh |-|      1 + tanh |-|          1 + tanh |-|          1 + tanh |-|           1 + tanh |-|       
 |                            \2/               \2/                 \2/               \2/                   \2/                   \2/                    \2/       
/
$$4\,\left({{x\,e^{2\,x}}\over{2\,e^{2\,x}+2}}-{{\log \left(e^{2\,x}+ 1\right)}\over{4}}\right)$$
Simplify
The graph
Plot the graph f(x)
The answer [src] 
                         2             /        2     \                                               2         /        2     \         2                        
        1            tanh (1/2)     log\1 + tanh (1/2)/   2*log(1 + tanh(1/2))    2*tanh(1/2)     tanh (1/2)*log\1 + tanh (1/2)/   2*tanh (1/2)*log(1 + tanh(1/2))
- -------------- - -------------- - ------------------- + -------------------- + -------------- - ------------------------------ + -------------------------------
          2                2                   2                     2                   2                        2                                 2             
  1 + tanh (1/2)   1 + tanh (1/2)      1 + tanh (1/2)        1 + tanh (1/2)      1 + tanh (1/2)           1 + tanh (1/2)                    1 + tanh (1/2)
$$\log 2-{{\left(e^2+1\right)\,\log \left(e^2+1\right)-2\,e^2}\over{e ^2+1}}$$ 
=
= 
                         2             /        2     \                                               2         /        2     \         2                        
        1            tanh (1/2)     log\1 + tanh (1/2)/   2*log(1 + tanh(1/2))    2*tanh(1/2)     tanh (1/2)*log\1 + tanh (1/2)/   2*tanh (1/2)*log(1 + tanh(1/2))
- -------------- - -------------- - ------------------- + -------------------- + -------------- - ------------------------------ + -------------------------------
          2                2                   2                     2                   2                        2                                 2             
  1 + tanh (1/2)   1 + tanh (1/2)      1 + tanh (1/2)        1 + tanh (1/2)      1 + tanh (1/2)           1 + tanh (1/2)                    1 + tanh (1/2)
$$- \frac{1}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tanh^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} - \frac{\log{\left(\tanh^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)} \tanh^{2}{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \log{\left(\tanh{\left(\frac{1}{2} \right)} + 1 \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1} + \frac{2 \tanh{\left(\frac{1}{2} \right)}}{\tanh^{2}{\left(\frac{1}{2} \right)} + 1}$$
Simplify
Numerical answer [src] 
0.327813325472738
0.327813325472738
The graph
Integral of x/(cosh(x))^2 dx

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

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