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pi-pi*(x-1+sqrt(2))/(2sqrt(2))+1/2sin((z+1+sqrt(2))180/sqrt(2))
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Solving integrals/ pi-pi*(x-1+sqrt(2))/(2sqrt(2))+1/2sin((z-1+sqrt(2))180/sqrt(2))

Integral of pi-pi*(x-1+sqrt(2))/(2sqrt(2))+1/2sin((z-1+sqrt(2))180/sqrt(2)) dx

The solution

You have entered [src]

  1                                                              
  /                                                              
 |                                                               
 |  /                             //          ___\       1  \\   
 |  |                          sin|\z - 1 + \/ 2 /*180*-----||   
 |  |        /          ___\      |                      ___||   
 |  |     pi*\x - 1 + \/ 2 /      \                    \/ 2 /|   
 |  |pi - ------------------ + ------------------------------| dx
 |  |              ___                       2               |   
 |  \          2*\/ 2                                        /   
 |                                                               
/                                                                
0

$$\int\limits_{0}^{1} \left(- \frac{\pi \left(x - 1 + \sqrt{2}\right)}{2 \sqrt{2}} + \frac{\sin{\left(\left(z - 1 + \sqrt{2}\right) 180 \cdot \frac{1}{\sqrt{2}} \right)}}{2} + \pi\right)\, dx$$

Integral(pi - pi*(x - 1*1 + sqrt(2))/(2*sqrt(2)) + sin((z - 1*1 + sqrt(2))*180/sqrt(2))/2, (x, 0, 1))

Detail solution

Integrate term-by-term:
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{\pi \left(x - 1 + \sqrt{2}\right)}{2 \sqrt{2}}\right)\, dx = - \int \frac{\pi \left(x - 1 + \sqrt{2}\right)}{2 \sqrt{2}}\, dx$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{\pi \left(x - 1 + \sqrt{2}\right)}{2 \sqrt{2}}\, dx = \frac{\sqrt{2} \pi \int \left(x - 1 + \sqrt{2}\right)\, dx}{4}$
    1. Integrate term-by-term:
      1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int x\, dx = \frac{x^{2}}{2}$
      1. The integral of a constant is the constant times the variable of integration:
        
        $\int \left(\left(-1\right) 1\right)\, dx = - x$
      1. The integral of a constant is the constant times the variable of integration:
        
        $\int \sqrt{2}\, dx = \sqrt{2} x$
      The result is: $\frac{x^{2}}{2} - x + \sqrt{2} x$
    So, the result is: $\frac{\sqrt{2} \pi \left(\frac{x^{2}}{2} - x + \sqrt{2} x\right)}{4}$
  So, the result is: $- \frac{\sqrt{2} \pi \left(\frac{x^{2}}{2} - x + \sqrt{2} x\right)}{4}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \frac{\sin{\left(\left(z - 1 + \sqrt{2}\right) 180 \cdot \frac{1}{\sqrt{2}} \right)}}{2}\, dx = \frac{x \sin{\left(\left(z - 1 + \sqrt{2}\right) 180 \cdot \frac{1}{\sqrt{2}} \right)}}{2}$
1. The integral of a constant is the constant times the variable of integration:
  
  $\int \pi\, dx = \pi x$
The result is: $\frac{x \sin{\left(\left(z - 1 + \sqrt{2}\right) 180 \cdot \frac{1}{\sqrt{2}} \right)}}{2} + \pi x - \frac{\sqrt{2} \pi \left(\frac{x^{2}}{2} - x + \sqrt{2} x\right)}{4}$
Now simplify:

$\frac{x \left(- \sqrt{2} \pi x + 4 \sin{\left(90 \sqrt{2} z - 90 \sqrt{2} + 180 \right)} + 2 \sqrt{2} \pi + 4 \pi\right)}{8}$
Add the constant of integration:

$\frac{x \left(- \sqrt{2} \pi x + 4 \sin{\left(90 \sqrt{2} z - 90 \sqrt{2} + 180 \right)} + 2 \sqrt{2} \pi + 4 \pi\right)}{8}+ \mathrm{constant}$

The answer is:

$\frac{x \left(- \sqrt{2} \pi x + 4 \sin{\left(90 \sqrt{2} z - 90 \sqrt{2} + 180 \right)} + 2 \sqrt{2} \pi + 4 \pi\right)}{8}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                                                                                         
 |                                                                                                                                          
 | /                             //          ___\       1  \\                      //          ___\       1  \            / 2              \
 | |                          sin|\z - 1 + \/ 2 /*180*-----||                 x*sin|\z - 1 + \/ 2 /*180*-----|        ___ |x            ___|
 | |        /          ___\      |                      ___||                      |                      ___|   pi*\/ 2 *|-- - x + x*\/ 2 |
 | |     pi*\x - 1 + \/ 2 /      \                    \/ 2 /|                      \                    \/ 2 /            \2               /
 | |pi - ------------------ + ------------------------------| dx = C + pi*x + -------------------------------- - ---------------------------
 | |              ___                       2               |                                2                                4             
 | \          2*\/ 2                                        /                                                                               
 |                                                                                                                                          
/

$$\int \left(- \frac{\pi \left(x - 1 + \sqrt{2}\right)}{2 \sqrt{2}} + \frac{\sin{\left(\left(z - 1 + \sqrt{2}\right) 180 \cdot \frac{1}{\sqrt{2}} \right)}}{2} + \pi\right)\, dx = C + \frac{x \sin{\left(\left(z - 1 + \sqrt{2}\right) 180 \cdot \frac{1}{\sqrt{2}} \right)}}{2} + \pi x - \frac{\sqrt{2} \pi \left(\frac{x^{2}}{2} - x + \sqrt{2} x\right)}{4}$$

Simplify

The answer [src]

        /           ___          ___\        ___
pi   sin\180 - 90*\/ 2  + 90*z*\/ 2 /   pi*\/ 2 
-- + -------------------------------- + --------
2                   2                      8

$$\frac{\sin{\left(90 \sqrt{2} z - 90 \sqrt{2} + 180 \right)}}{2} + \frac{\sqrt{2} \pi}{8} + \frac{\pi}{2}$$

        /           ___          ___\        ___
pi   sin\180 - 90*\/ 2  + 90*z*\/ 2 /   pi*\/ 2 
-- + -------------------------------- + --------
2                   2                      8

$$\frac{\sin{\left(90 \sqrt{2} z - 90 \sqrt{2} + 180 \right)}}{2} + \frac{\sqrt{2} \pi}{8} + \frac{\pi}{2}$$

Упростить

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

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Integral of pi-pi*(x-1+sqrt(2))/(2sqrt(2))+1/2sin((z-1+sqrt(2))180/sqrt(2)) dx

Limits of integration:

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Piecewise:

The solution