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Identical expressions
(z^ two)*ch(two *z)+ four *(z^ four)-z/ four -sh(six *pi*z)
(z squared ) multiply by ch(2 multiply by z) plus 4 multiply by (z to the power of 4) minus z divide by 4 minus sh(6 multiply by Pi multiply by z)
(z to the power of two) multiply by ch(two multiply by z) plus four multiply by (z to the power of four) minus z divide by four minus sh(six multiply by Pi multiply by z)
(z2)*ch(2*z)+4*(z4)-z/4-sh(6*pi*z)
z2*ch2*z+4*z4-z/4-sh6*pi*z
(z²)*ch(2*z)+4*(z⁴)-z/4-sh(6*pi*z)
(z to the power of 2)*ch(2*z)+4*(z to the power of 4)-z/4-sh(6*pi*z)
(z^2)ch(2z)+4(z^4)-z/4-sh(6piz)
(z2)ch(2z)+4(z4)-z/4-sh(6piz)
z2ch2z+4z4-z/4-sh6piz
z^2ch2z+4z^4-z/4-sh6piz
(z^2)*ch(2*z)+4*(z^4)-z divide by 4-sh(6*pi*z)
Similar expressions
(z^2)*ch(2*z)+4*(z^4)-z/4+sh(6*pi*z)
(z^2)*ch(2*z)-4*(z^4)-z/4-sh(6*pi*z)
(z^2)*ch(2*z)+4*(z^4)+z/4-sh(6*pi*z)

Solving integrals/ (z^2)*ch(2*z)+4*(z^4)-z/4-sh(6*pi*z)

Integral of (z^2)ch(2z)+4(z^4)-z/4-sh(6pi*z) dz

The solution

You have entered [src]

 -1 + 3*I                                           
     /                                              
    |                                               
    |    / 2                4   z               \   
    |    |z *cosh(2*z) + 4*z  - - - sinh(6*pi*z)| dz
    |    \                      4               /   
    |                                               
   /                                                
 2 - 2*I

$$\int\limits_{2 - 2 i}^{-1 + 3 i} \left(\left(- \frac{z}{4} + \left(4 z^{4} + z^{2} \cosh{\left(2 z \right)}\right)\right) - \sinh{\left(6 \pi z \right)}\right)\, dz$$

Integral(z^2*cosh(2*z) + 4*z^4 - z/4 - sinh((6*pi)*z), (z, 2 - 2*i, -1 + 3*i))

Detail solution

Integrate term-by-term:
1. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{z}{4}\right)\, dz = - \frac{\int z\, dz}{4}$
    1. The integral of $z^{n}$ is $\frac{z^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int z\, dz = \frac{z^{2}}{2}$
    So, the result is: $- \frac{z^{2}}{8}$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 4 z^{4}\, dz = 4 \int z^{4}\, dz$
      1. The integral of $z^{n}$ is $\frac{z^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int z^{4}\, dz = \frac{z^{5}}{5}$
      So, the result is: $\frac{4 z^{5}}{5}$
    1. Don't know the steps in finding this integral.
      
      But the integral is
      
      $\frac{z^{2} \sinh{\left(2 z \right)}}{2} - \frac{z \cosh{\left(2 z \right)}}{2} + \frac{\sinh{\left(2 z \right)}}{4}$
    The result is: $\frac{4 z^{5}}{5} + \frac{z^{2} \sinh{\left(2 z \right)}}{2} - \frac{z \cosh{\left(2 z \right)}}{2} + \frac{\sinh{\left(2 z \right)}}{4}$
  The result is: $\frac{4 z^{5}}{5} + \frac{z^{2} \sinh{\left(2 z \right)}}{2} - \frac{z^{2}}{8} - \frac{z \cosh{\left(2 z \right)}}{2} + \frac{\sinh{\left(2 z \right)}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \sinh{\left(6 \pi z \right)}\right)\, dz = - \int \sinh{\left(6 \pi z \right)}\, dz$
  1. Don't know the steps in finding this integral.
    
    But the integral is
    
    $- \frac{1}{3 \pi \tanh^{2}{\left(3 \pi z \right)} - 3 \pi}$
  So, the result is: $\frac{1}{3 \pi \tanh^{2}{\left(3 \pi z \right)} - 3 \pi}$
The result is: $\frac{4 z^{5}}{5} + \frac{z^{2} \sinh{\left(2 z \right)}}{2} - \frac{z^{2}}{8} - \frac{z \cosh{\left(2 z \right)}}{2} + \frac{\sinh{\left(2 z \right)}}{4} + \frac{1}{3 \pi \tanh^{2}{\left(3 \pi z \right)} - 3 \pi}$
Now simplify:

$\frac{4 z^{5}}{5} + \frac{z^{2} \sinh{\left(2 z \right)}}{2} - \frac{z^{2}}{8} - \frac{z \cosh{\left(2 z \right)}}{2} + \frac{\sinh{\left(2 z \right)}}{4} - \frac{\cosh^{2}{\left(3 \pi z \right)}}{3 \pi}$
Add the constant of integration:

$\frac{4 z^{5}}{5} + \frac{z^{2} \sinh{\left(2 z \right)}}{2} - \frac{z^{2}}{8} - \frac{z \cosh{\left(2 z \right)}}{2} + \frac{\sinh{\left(2 z \right)}}{4} - \frac{\cosh^{2}{\left(3 \pi z \right)}}{3 \pi}+ \mathrm{constant}$

The answer is:

$\frac{4 z^{5}}{5} + \frac{z^{2} \sinh{\left(2 z \right)}}{2} - \frac{z^{2}}{8} - \frac{z \cosh{\left(2 z \right)}}{2} + \frac{\sinh{\left(2 z \right)}}{4} - \frac{\cosh^{2}{\left(3 \pi z \right)}}{3 \pi}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                 
 |                                                                                 2                  5    2                        
 | / 2                4   z               \                      1                z    sinh(2*z)   4*z    z *sinh(2*z)   z*cosh(2*z)
 | |z *cosh(2*z) + 4*z  - - - sinh(6*pi*z)| dz = C + -------------------------- - -- + --------- + ---- + ------------ - -----------
 | \                      4               /                           2           8        4        5          2              2     
 |                                                   -3*pi + 3*pi*tanh (3*pi*z)                                                     
/

$$\int \left(\left(- \frac{z}{4} + \left(4 z^{4} + z^{2} \cosh{\left(2 z \right)}\right)\right) - \sinh{\left(6 \pi z \right)}\right)\, dz = C + \frac{4 z^{5}}{5} + \frac{z^{2} \sinh{\left(2 z \right)}}{2} - \frac{z^{2}}{8} - \frac{z \cosh{\left(2 z \right)}}{2} + \frac{\sinh{\left(2 z \right)}}{4} + \frac{1}{3 \pi \tanh^{2}{\left(3 \pi z \right)} - 3 \pi}$$

Simplify

The answer [src]

             5                             2                             2               5             2                                                     2                                                                                         
  4*(2 - 2*I)    sinh(4 - 4*I)   (-1 + 3*I)    sinh(-2 + 6*I)   (2 - 2*I)    4*(-1 + 3*I)    (-1 + 3*I) *sinh(-2 + 6*I)   (2 - 2*I)*cosh(4 - 4*I)   (2 - 2*I) *sinh(4 - 4*I)   (-1 + 3*I)*cosh(-2 + 6*I)   cosh(6*pi*(-1 + 3*I))   cosh(6*pi*(2 - 2*I))
- ------------ - ------------- - ----------- + -------------- + ---------- + ------------- + -------------------------- + ----------------------- - ------------------------ - ------------------------- - --------------------- + --------------------
       5               4              8              4              8              5                     2                           2                         2                           2                        6*pi                   6*pi

$$- \frac{4 \left(2 - 2 i\right)^{5}}{5} - \frac{\left(2 - 2 i\right)^{2} \sinh{\left(4 - 4 i \right)}}{2} + \frac{4 \left(-1 + 3 i\right)^{5}}{5} - \frac{\sinh{\left(4 - 4 i \right)}}{4} - \frac{\left(-1 + 3 i\right) \cosh{\left(-2 + 6 i \right)}}{2} + \frac{\left(2 - 2 i\right)^{2}}{8} + \frac{\sinh{\left(-2 + 6 i \right)}}{4} - \frac{\cosh{\left(6 \pi \left(-1 + 3 i\right) \right)}}{6 \pi} + \frac{\cosh{\left(6 \pi \left(2 - 2 i\right) \right)}}{6 \pi} - \frac{\left(-1 + 3 i\right)^{2}}{8} + \frac{\left(-1 + 3 i\right)^{2} \sinh{\left(-2 + 6 i \right)}}{2} + \frac{\left(2 - 2 i\right) \cosh{\left(4 - 4 i \right)}}{2}$$

             5                             2                             2               5             2                                                     2                                                                                         
  4*(2 - 2*I)    sinh(4 - 4*I)   (-1 + 3*I)    sinh(-2 + 6*I)   (2 - 2*I)    4*(-1 + 3*I)    (-1 + 3*I) *sinh(-2 + 6*I)   (2 - 2*I)*cosh(4 - 4*I)   (2 - 2*I) *sinh(4 - 4*I)   (-1 + 3*I)*cosh(-2 + 6*I)   cosh(6*pi*(-1 + 3*I))   cosh(6*pi*(2 - 2*I))
- ------------ - ------------- - ----------- + -------------- + ---------- + ------------- + -------------------------- + ----------------------- - ------------------------ - ------------------------- - --------------------- + --------------------
       5               4              8              4              8              5                     2                           2                         2                           2                        6*pi                   6*pi

-4*(2 - 2*i)^5/5 - sinh(4 - 4*i)/4 - (-1 + 3*i)^2/8 + sinh(-2 + 6*i)/4 + (2 - 2*i)^2/8 + 4*(-1 + 3*i)^5/5 + (-1 + 3*i)^2*sinh(-2 + 6*i)/2 + (2 - 2*i)*cosh(4 - 4*i)/2 - (2 - 2*i)^2*sinh(4 - 4*i)/2 - (-1 + 3*i)*cosh(-2 + 6*i)/2 - cosh(6*pi*(-1 + 3*i))/(6*pi) + cosh(6*pi*(2 - 2*i))/(6*pi)

Numerical answer [src]

(625439238811323.0 - 140.783774074311j)

(625439238811323.0 - 140.783774074311j)

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

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Identical expressions

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Integral of (z^2)ch(2z)+4(z^4)-z/4-sh(6pi*z) dz

Limits of integration:

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (z^2)*ch(2*z)+4*(z^4)-z/4-sh(6*pi*z) dz

Limits of integration:

The graph:

Piecewise:

The solution

Integral of (z^2)ch(2z)+4(z^4)-z/4-sh(6pi*z) dz